Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
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Solve the equation $9^x+3^{2x-1}=2^{x+3.5}+2^{x+0.5}$ Solve the equation $$9^x+3^{2x-1}=2^{x+3.5}+2^{x+0.5}$$
The equation is equivalent to $$3^{2x}+3^{2x}\cdot3^{-1}=2^x\cdot2^{3.5}+2^x\cdot2^{0.5}$$ which is $$3^{2x}\left(1+\dfrac{1}{3}\right)=2^x\left(2^\frac72+2^\frac12\right)$$ We can write the last equation as $$... | you just need to simplify the last expression as
$\displaystyle \left(\frac{9}{2}\right)^{x} \ =\ \frac{27\sqrt{2}}{4} \ =\ \left(\frac{3^{3}}{2^{( 3/2)}}\right) \ =\ \left(\frac{9^{1.5}}{2^{1.5}}\right)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4618792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Composing function The function is $f: \Bbb{R}\rightarrow\Bbb{R}$ defined as $f(x)= 2/ (x -3).$
I need to find $(f o f)(1).$
I would like to ask which of the following answers are the right one for writing this function.
$( f o f) ( 1 ) = ( f ( f ( 1 ) ) )= ( f ( 2/ 1 - 3) = ( 2 / 1 - 3 - 2 ) = -1/2 $
or
$( f o f ) ( 1... | At first note that we should consider a different domain of the function $f$:
\begin{align*}
&f:\mathbb{R}\setminus\{3\}\to\mathbb{R}\\
&f(x)=\frac{2}{x-3}
\end{align*}
since $f$ is not defined at $x=3$.
We can calculate $(f\circ f)(1)=f(f(1))$ as follows:
\begin{align*}
\color{blue}{f(f(1))}=
\begin{cases}
\frac{2}{f(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4618947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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Parametrizations of $ x^4+y^4+z^4=9t^2 $ integer solutions I would like to derive all the parametrizations for the nontrivial solutions of this Diophantine equation:
$ x^4+y^4+z^4=9t^2 $
I already know that with the Fauquembergue's parametrization I can find infinite solutions (essentially multiplying by 3 the terms in... | The Diophantine equation,
$$x^4+y^4+z^4 = nt^2$$
has quite an interesting history, being studied by Fauquembergue, Proth, Ramanujan, etc.
I. Case n = 1
As alluded to by the OP, Fauquembergue found that if $a^2+b^2 = c^2$, then,
$$(ab)^4+(ac)^4+(bc)^4 = (a^4+a^2b^2+b^4)^2$$
So using the simplest Pythagorean triple $(3,4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4621481",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Prove that $\frac{x^2}{x^2 + y^2}$ is irreducible in $\Bbb{Q}[x,y,\frac{1}{x^2+y^2}]_0$ I was doing Vakil's FOAG, in exercise 5.4 N needs to investigate an important example that $\Bbb{Q}[x,y,\frac{1}{x^2+y^2}]_0$ is not UFD (where we take localization of $\Bbb{Q}[x,y]$ at $x^2+y^2$ and take the degree zero part of it)... | We have $p / (x^2 + y^2)^m * q / (x^2 + y^2)^n = x^2 / (x^2 + y^2)$. Rearranging, we get $pq = x^2 (x^2 + y^2)^{m+n-1}$. Assume that neither $p$ nor $q$ are divisible by $x^2+y^2$. Note that as $p / (x^2 + y^2)^m $ has degree 0 and $deg(p) \geq 0$, $m$ cannot be negative, and similarly for $n$.
Now, $\mathbb{Q}[x,y]$ i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the probability that the sum of 6 four-sided dice is less than or equal to 14? The solution given is shown below.
My question is how did they count the numerator like that?
What is the explanation for it please?
$$\begin{align}
\frac{C_6^{14}-C_1^6\times C_6^{10}+C_2^6 \times C_6^6}{4^6}&=\frac{3003-6\times 210... | (I'm assuming you're familiar with Stars-and-Bars. If not, see true blue anil's solution which deals with that part of it.)
Consider finding the sum of 6 positive integers that is at most $14$, which is equivalent to 7 non-negative integers that sum to exactly $14-6=8$ (where the last variable is the dummy variable for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4627424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Solving a first order ordinary differential equation with no initial conditions
Solve the following differential equation. $$ \dfrac{dy}{dx} = \dfrac{3y - 3x}{2x+3y} $$
Answer:
Let $y = xv$. Then we have:
\begin{align*}
\dfrac{dy}{dx} &= x \dfrac{dv}{dx} + v \\
v &= \dfrac{ y}{x} \\
\dfrac{dy}{dx} &=
\dfrac{ 3\lef... | Notice that
$$y=(2x+3y)(x+c)\iff y=\frac{2x(x+c)}{-3(x+c)+1},$$when $(x+c)\not=\frac{1}{3}$. Setting $c=0$ and find $dy/dx$ we have $$\frac{dy}{dx}=\frac{2(2-3x)x}{(3x-1)^2}\not=\frac{3(\frac{2x^2}{1-3x})-3x}{2x+3(\frac{2x^2}{1-3x})}.$$
So your function $y$ is not solution of the ode.
Instead, your first way is correct... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4627965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding the total number of objects(stones). I have $N$ stones. Then the stones are arranged in ascending order of weights. If I remove three stones that are heaviest, then the total weight of the stones decreases by $35$%. Now if I remove the three lightest stones, the total weight of the stones further decreases by a... | For another approach: The three heaviest stones weigh $\frac{7}{20} S$. The three lightest stones weigh $\frac{5}{20} S$. The $N-6$ other stones weigh $\frac{8}{20} S$. The averages of these three groups must be in order:
$$ \frac{5 S}{3 \cdot 20} < \frac{8 S}{(N-6)20} < \frac{7 S}{3 \cdot 20} $$
$$ \frac{24}{5} > N-6 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4629953",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Solve for all $x$ such that $x^3 = 2x + 1, x^4 = 3x + 2, x^5 = 5x + 3, x^6 = 8x +5 \cdots$ Question:
Solve for all $x$ such that $\begin{cases}&{x}^{3}=2x+1\\&{x}^{4}=3x+2\\&{x}^{5}=5x+3\\&{x}^{6}=8x+5\\&\vdots\end{cases}$.
My attempt:
I sum up everything.
$$\begin{aligned}\sum_{i=1}^n x^{i+2} &= (2 + 3 + 5 + 8 + \cdo... | $x^3 = 2x + 1$ has an evident rational solution, that being $-1.$ Next, $x+1$ divides $x^3 - 2x - 1,$ the quotient is $x^2 - x-1$ for
$$ x^3 - 2x - 1 = (x+1) (x^2 - x - 1) $$
However $-1$ does not satisfy $x^4 = 3x + 2.$
We are left with those numbers $x$ with $x^2 = x+1.$ We can find the values of $x^n$ by a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4632276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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Check an $\epsilon-\delta$ proof of $\lim_{ (x,y) \to (3,1)} \frac{x}{y} = 3$ I am going to prove
$$\lim_{ (x,y) \to (3,1)} \frac{x}{y} = 3$$
by using $\epsilon-\delta$ argument.
My Attempts
Preliminary Analysis
For every $\epsilon >0$, we need to find a suitable $\delta$ such that:
If $\Vert (x,y)-(3,1)\Vert _2 < \del... | Your proof is not correct because
$\left|\dfrac xy-3\right|=\left|\dfrac xy-3y+3y-3\right|\leqslant\left|\dfrac1y\right||x-3|+3|y-1|$
is wrong (see my comment).
So you could proceed as follows:
$\begin{align}\left|\dfrac xy-3\right|&=\left|\dfrac 1y\right|\,\big|x-3y\big|=\left|\dfrac 1y\right|\,\big|x-3-3\big(y-1\big)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4633379",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Divergence theorem example I have this problem:
Verify the divergence theorem $$
\oint_{S} \mathbf{A} \cdot d \mathbf{S}=\int_{v} \nabla \cdot \mathbf{A} d v$$ for the following case: $\mathbf{A}=2 \rho z \mathbf{a}_{\rho}+3 z \sin \phi \mathbf{a}_{\phi}-4 \rho \cos \phi \mathbf{a}_{z}$ and $S$ is the surface of the w... | On the $z = 0$ surface, the normal vector is $-\hat{z}$, not $\hat{z}$. That means that you should have
\begin{eqnarray}
&=&\iint_{\rho=2}2 \rho^2 z d\phi dz +\iint_{\phi=\pi/4} 3 z \sin \phi d\rho dz\\
& &\ \ \ \ \color{red}{-} \iint_{z=0}-4 \rho^2 \cos \phi d\phi d\rho+\iint_{z=5}-4 \rho^2 \cos \phi d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4633813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Number of distinct arrangement of $(a,b,c,d,e)$
If $a<b<c<d<e $ be
positive integer such that
$a+b+c+d+e=20$.
Then number of distinct
arrangement of $(a,b,c,d,e)$ is
Here the largest value of $e$ is $10$
like $a\ b\ c\ d\ e$
as $ \ \ 1\ 2\ 3\ 4\ 10$
And least value is $6$
like $ a\ b\ c\ d\ e$
as $\ \ 2\ 3\... | Equivalent problem; find non-negative integers $p\leq q\leq r\leq s\leq t$ such that $p+q+r+s+t=5$. The 5 numbers you want will be given by
$$
\begin{aligned}
a &= 1+p \\
b &= 2+q \\
c &= 3+r \\
d &= 4+s \\
e &= 5+t
\end{aligned}
$$
Repeated stars and bars and division by number of permutations will give you that expre... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4634663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Create unique table rotation for an event I am planning to host an event with 14 people participating in it. There will be five tables in total. (4 tables with 3 people each and 1 with 2).
I want to rotate the tables 5-6 times and to ensure that everyone gets to meet as many people as possible and minimize (if not comp... | The best that I could come up with was this seven round schedule. Out of the $14\times 13/2=91$ pairs of people, this schedule causes all but six pairs of people to meet each other, with only six pairs of people meeting twice.
Table 1
Table 2
Table 3
Table 4
Table 5
Round 1
$4, 6, 7$
$1, 2, 5$
$9, 12, 14$
$3, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4635982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Given a triangle $\triangle ABC$, with an internal point $K$, find $\angle AKC$. As title suggests, the question is to find the measure of $\angle AKC$ in the triangle $\triangle ABC$, with an internal point $K$ and some given angles.
This is a pretty challenging problem, I'm going to post my solution below as an answ... | We have $\angle ABC = 66^\circ$ and $\angle AKC = 150^\circ$. Assume $BC = 1$, then $AC = \frac{\sin 48^\circ}{\sin 66^\circ} = \frac{\sin 48^\circ}{\cos 24^\circ} = 2 \sin 24^\circ$ and $KC = \frac{\sin 12^\circ}{\sin 150^\circ} = 2\sin 12^\circ$.
Now
$$AK^2 = AC^2 + KC^2 - 2AC \cdot KC \cdot \cos \angle ACK = 4\sin^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4637245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Mysterious sum equal to $\frac{7(p^2-1)}{24}$ where $p \equiv 1 \pmod{4}$ Consider a prime number $p \equiv 1 \pmod{4}$ and $n_p$ denotes the remainder of $n$ upon division by $p$. Let $A_p=\{ a \in [[0,p]] \mid {(a+1)^2}_p<{a^2}_p\}$.
I Conjecture
$$\sum_{a \in A_p } a=\dfrac{7(p^2-1)}{24}$$
**Addition, a simpler way ... | I will leave several details for you to check, ask me if there are any issues with verifying them.
Let $$A_p=\{ a \in [[0,p-1]] \mid {(a+1)^2}_p<{a^2}_p\},$$
$$B_p=\{ a \in [[0,p-1]] \mid {(a+1)^2}_p>{a^2}_p\},$$
$$C_p=\{ a \in [[0,p-1]] \mid {(a+1)^2}_p={a^2}_p\}.$$
Then $A_p \cup B_p \cup C_p = [[0,p-1]]$. Check that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4639883",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
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"answer_id": 0
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Find limit of sequence $(x_n)$ Find limit of sequence $(x_n)$:
$$x_1 = a >0$$
$$x_{n+1} = \frac{n}{2n-1}\frac{x_n^2+2}{x_n}, n \in Z^+$$
I think I can prove $(x_n)$ is low bounded (which is obvious that $x_n>0$) and decreasing sequence. Then I can calculate the limit of sequence is $\sqrt{2}$
All my attempts to prove i... | A recursion shows that the $x_n$ for $n \ge 2$ are well-defined and bounded below by $\sqrt{2}$, since $x^2+2-2\sqrt{x} = (x-\sqrt{2})^2$ is always positive. Set $y_n = x_n-\sqrt{2}$. Then for $n \ge 2$,
\begin{eqnarray*}
y_{n+1}
&=& \frac{n(x_n^2+2)-\sqrt{2}(2n-1)x_n}{(2n-1)x_n} \\
&=& \frac{n(x_n-\sqrt{2})^2+\sqrt{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4641083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Limit of $(|x| + |y|)\ln(x^2 + y^4)$ at $(0,0)$ I want to show that $$\lim\limits_{(x,y) \to (0,0)} (\lvert x \rvert + \lvert y \rvert)\ln(x^2 + y^4) = 0$$
First I let $\lVert (x,y) \rVert = \lvert x \rvert + \lvert y \rvert\ < \delta$, and assume that $x,y < 1$ so $x^2 + y^4 < \lvert x \rvert + \lvert y \rvert$. Then ... | We have that by generalized mean inequality
$$\frac{\lvert x \rvert + \lvert y \rvert}2 \le \sqrt[4]{\frac{x^4 +y^4}2} \iff \lvert x \rvert + \lvert y \rvert\le\frac 2{\sqrt[4]2}\sqrt[4]{x^4 +y^4}$$
then assuming wlog $x^2 + y^4<1$
$$(\lvert x \rvert + \lvert y \rvert)\left|\ln(x^2 + y^4)\right|
\le \frac 2{\sqrt[4]2}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4641244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Solve the differential equation: $(x^2-y^2)dx+2xydy=0$. Given $(x^2-y^2)dx+2xydy=0$
My solution-
Divide the differential equation by $dx$
$\Rightarrow x^2-y^2+2xy\frac{dy}{dx}=0$
$\Rightarrow 2xy\frac{dy}{dx}=y^2-x^2$
Divide both sides by $2xy$
$\Rightarrow \frac{dy}{dx}=\frac{1}{2}[\frac{y}{x}-\frac{x}{y}]$
This is a ... | Another approach let's call $u=x^2+y^2$.
Then $xu'=2x^2+2xyy'=2x^2+(y^2-x^2)=x^2+y^2=u$
Which solves to $u=cx\iff x^2+y^2=cx$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4642892",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Evaluate: $\int \frac{2}{(1-x)(1+x^2)}dx$ Given $\int \frac{2}{(1-x)(1+x^2)}dx$
The most obvious approach is to use Partial fractions
Let $\frac{2}{(1-x)(1+x^2)}=\frac{A}{1-x}+\frac{Bx+C}{1+x^2}$
$\Rightarrow \frac{2}{(1-x)(1+x^2)}=\frac{A+Ax^2+Bx-Bx^2+C-Cx}{(1-x)(1+x^2)}$
We get $A=1, B=1, C=1$
The integral now become... | Here is another way to integrate
\begin{align}
&\int \frac{2}{(1-x)(1+x^2)}dx\\
=& \int \frac{2(1+x)}{1-x^4}dx= \int \frac{2x}{1-x^4}+ \frac{2}{1-x^4} \ dx\\
=& \int \frac{1}{1-x^4} \ d(x^2)+\int \frac{1}{1-x^2}+\frac{1}{1+x^2}\ dx\\
= &\ \frac12\ln|\frac{1+x^2}{1-x^2}|+ \frac12\ln|\frac{1+x}{1-x}|+\tan^{-1}x+C
\end{al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4643054",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Show the value of $\mathbb{E}(X^2Y^2)$ Suppose that $(X,Y)$ is distributed ranodm 2-vectors having the normal distribution with $\mathbb{E}X=\mathbb{E}Y=0,\mathbb{Var}(X)=\mathbb{Var}(Y)=1,$ and $\text{Cov}(X,Y)=\theta\in(-1,1).$ Show the value of $\mathbb{E}(X^2Y^2).$
The joint pdf of $(X,Y)$ is $$p(x,y)=\frac{1}{2\... | $$\begin{align*}
\Bbb E\left[X^2Y^2\right] &= \frac1{2\pi \sqrt{1-\theta^2}} \int\limits_0^\infty \int\limits_0^\infty \left(u^2-v^2\right)^2 e^{\Large-\frac{(1-\theta) u^2+(1+\theta)v^2}{2(1-\theta^2)}} \, du \, dv \\
\tag{1} &= \frac4\pi \int\limits_0^\infty \int\limits_0^\infty \left((1+\theta)s^2-(1-\theta)t^2\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4643345",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Linear Regression of Quadratic Points I was casually experimenting with Desmos's linear regression and noticed some patterns in the best fit line for points which fit y = x^2.
For (0,0) and (1,1) the best fit line is y = 1x+0 and r = 1 (the correlation coefficient).
For (0,0) , (1,1) , (2,4) , ... , (10,100) the best f... | What is the formula for the line of best fit here?
In this case, because the points we take follow an algebraic pattern, we can calculate a linear regression on the points $\{(0,0), (1,1), (2,4), \dots, (n,n^2)\}$ in terms of $n$, and see what happens.
To do this, we start with the overconstrained system of equations t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4644998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Compute the integral $\int_{-\infty}^\infty \frac{\sin(x) (1+\cos(x))}{x(2+\cos(x))} dx$ Background
I recently found out about Lobachevsky's integral formula, so I tried to create a problem on my own for which I'd be able to apply this formula. The problem is presented below.
Problem
Compute the integral $\int_{-\inft... | Another approach:
Using the Fourier series $$1 + 2 \sum_{k=1}^{\infty} (-1)^{n} a^{n}\cos(nx) = \frac{1-a^{2}}{1+2a \cos (x) +a^{2}}, \quad |a| <1, $$ we have
$$ \begin{align} \int_{-\infty}^{\infty} \frac{\sin (x)}{x}\frac{1-a^{2}}{1+2a \cos (x) +a^{2}} \, \mathrm dx &= \int_{-\infty}^{\infty} \frac{\sin (x)}{x}\left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4645216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 0
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Proof by induction of inequality. For $n \geq 2, 3^n \gt 2^n+n^2$ I'm trying to understand proof by induction on inequalities.
One problem I have is:
Prove that for $n \geq 2$ $3^n \gt 2^n+n^2$
I can prove the base case as $n=2$:
$3^2 \gt 2^2+2^2$
$9 \gt 8$
What I don't understand is the inductive step. How do I ... | Suppose $3^k > 2^k + k^2$ you have to show $3^{k+1} > 2^{k+1} + (k+1)^2$. From the inductive step, $3^{k+1} = 3\cdot 3^k > 3(2^k+k^2)= 2^{k+1}+2^k+k^2+2k^2>2^{k+1}+k^2+2k+1=2^{k+1}+(k+1)^2$. This is true because $2^k \ge 2k$, and $2k^2 > 1$. Thus by induction, we are done.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4645394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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For what $c \in \mathbb{Q}$ is $\sqrt{2}+c\sqrt{3}$ a primitive element of $\mathbb{Q}(\sqrt{2},\sqrt{3})$? I am trying to find the $c\in\mathbb{Q}$ so that $\sqrt{2}+c\sqrt{3}$ is a primitive element of $\mathbb{Q}(\sqrt{2},\sqrt{3})$, i.e. $\mathbb{Q}(\sqrt{2}+c\sqrt{3})=\mathbb{Q}(\sqrt{2},\sqrt{3})$.
In class we h... | We need to decide whether $\theta=\sqrt{2}+c\sqrt{3}$ has degree $4$ over $\mathbb Q$.
This reduces to deciding whether $1,\theta,\theta^2,\theta^3$ are linearly independent over $\mathbb Q$.
We have
$$
\pmatrix{1 \\ \theta \\ \theta^2 \\ \theta^3}
=
\pmatrix{
1 & 0 & 0 & 0 \\
0 & 1 & c & 0 \\
2+3c^2 & 0 & 0 & 2c \\
0 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4645805",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Solving inequality $\frac{7x+12}{x} \geq 3$ I was sitting with my friend in a park, when he thought of and asked for solving this inequality:
$$
\frac{7x+12}{x} \geq 3
$$
I was confident enough that I could solve this example as continued:
$$
7x + 12 \geq 3x
$$
$$
4x \geq -12
$$
$$
x \geq -3
$$
He simply told me to put... | You made a mistake when you multiplied both sides of the inequality by $x$. Since multiplying an inequality by a negative number reverses the direction of the inequality, you have to consider cases when you multiply both sides of the inequality by $x$.
Method 1
Case 1: $x > 0$
\begin{align*}
\frac{7x + 12}{x} & \geq ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4646839",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
How many methods are there to evaluate $\int_0^{\infty} \frac{1}{\left(x+\frac{1}{x}\right)^{2n}}$? Background
As I had found the integral
$$I=\int_0^{\infty} \frac{1}{\left(x+\frac{1}{x}\right)^2} d x =\frac{\pi}{4}, $$
by using $x\mapsto \frac{1}{x}$ yields
$\displaystyle I=\int_0^{\infty} \frac{\frac{1}{x^2}}{\left(... | $$
\begin{align}
\int_{0}^{\infty}\frac{1}{\left(x+\frac{1}{x}\right)^{2n}}dx &= \int_{0}^{\infty}\frac{x^{2n}}{\left(1+x^{2}\right)^{2n}}dx \\
&= \frac{1}{2}\int_{-\infty}^{\infty}\frac{x^{2n}}{\left(1+x^{2}\right)^{2n}}dx \\
&= \frac{1}{2}\int_{-\infty}^{\infty}\frac{x^{2n}}{\left(x-i\right)^{2n}\left(x+i\right)^{2n}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4647346",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
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Define $f\left(x\right) = \frac{x}{x + \frac{x}{x + \frac{x}{x + \frac{x}{x + \vdots}}}}$. What is $f'\left(x\right)$, the derivative? Since $f\left(x\right)$ is indeed an infinitely deep continued fraction, I have seen that $f\left(x\right) = \frac{x}{x + f\left(x\right)}$, but taking the derivative of both sides from... | We have $f^2(x) + xf(x) - x = 0$ equation, there are two functions satisfying this equation:
$$f_+(x) = \frac{-x+\sqrt{x^2+4x}}{2}$$ and
$$f_-(x) = \frac{-x - \sqrt{x^2+4x}}{2}$$
both functions are undefined for $ -4 < x < 0$. There are also the following properties for these functions:
$$x > 0 \implies f_+(x) > 0, f_-... | {
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"source": "stackexchange",
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Proof that $n^3+2n$ is divisible by $3$ I'm trying to freshen up for school in another month, and I'm struggling with the simplest of proofs!
Problem:
For any natural number $n , n^3 + 2n$ is divisible by $3.$
This makes sense
Proof:
Basis Step: If $n = 0,$ then $n^3 + 2n = 0^3 +$
$2 \times 0 = 0.$ So it is divisi... | $$
\begin{align}
n^3+2n &= n(n^2+2) \\
&= n(n^2−1+3)\\
&= n((n^2−1^2)+3)\\
&= n((n−1)(n+1)+3)\\
&= n(n−1)(n+1)+3n
\end{align}
$$
The key is to identify the factor $(n^2-1^2)=(n-1)(n+1)$. Looking at the result you can tell that it must be divisable by 3 because one of $n$, $(n-1)$ or $(n+1)$ ... | {
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"source": "stackexchange",
"question_score": "16",
"answer_count": 13,
"answer_id": 3
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Proof that $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$ Why is $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$ $\space$ ?
| Gathering as many proofs as we can? Write the series recursively:
$$S(n) = S(n - 1) + n \tag{1}$$
Substitute $n \to n+1$ :
$$S(n + 1) = S(n) + n + 1\tag{2}$$
Equation (2) subtract Equation (1):
$$S(n+1) - S(n) = S(n) + 1 - S(n - 1) \tag{3}$$
And write it up:
$$\begin{cases}
S(n+1) &= 2S(n) -S(n-1) + 1 \\
S(n) &= S(n) ... | {
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"source": "stackexchange",
"question_score": "136",
"answer_count": 36,
"answer_id": 24
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Proving that this sum $\sum\limits_{0 < k < \frac{2p}{3}} { p \choose k}$ is divisible by $p^{2}$ How does one prove that for a prime $p \geq 5$ the sum : $$\sum\limits_{0 < k < \frac{2p}{3}} { p \choose k}$$ is divisible by $p^{2}$.
Since each term of $\displaystyle \sum\limits_{0 < k < \frac{2p}{3}} { p \choose k}$ ... | Since we are working in the field $\mathbb{F}_p$ we can write
$$\frac{(p-1)(p-2) \cdots (p-k+1)}{1 \cdot 2 \cdots k}$$ as
$$\frac{(-1)(-2) \cdots (-(k-1))}{1 \cdot2 \cdots k}$$
= $$\frac{(-1)^{k-1}}{k}$$
Let $N = [\frac{2p}{3}]$ and $M = [\frac{N}{2}]$
Thus what we need is
$$ \sum_{k=1}^{N} \frac{(-1)^{k-1}}{k}$$ $$ ... | {
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"url": "https://math.stackexchange.com/questions/3925",
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"question_score": "10",
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"answer_id": 0
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Proving a binomial sum identity $\sum _{k=0}^n \binom nk \frac{(-1)^k}{2k+1} = \frac{(2n)!!}{(2n+1)!!}$ Mathematica tells me that
$$\sum _{k=0}^n { n \choose k} \frac{(-1)^k}{2k+1} = \frac{(2n)!!}{(2n+1)!!}.$$
Although I have not been able to come up with a proof.
Proofs, hints, or references are all welcome.
| In trying to prove
$$S_n = \sum_{k=0}^n \frac{(-1)^k}{2k+1} {n\choose k}
= \frac{(2n)!!}{(2n+1)!!}$$
we introduce
$$f(z) = n! (-1)^n \frac{1}{2z+1} \prod_{q=0}^n \frac{1}{z-q}.$$
This has the property that for $0\le k\le n$
$$\mathrm{Res}_{z=k} f(z)
= n! (-1)^n \frac{1}{2k+1}
\prod_{q=0}^{k-1} \frac{1}{k-q}
\prod_{q=k+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 8,
"answer_id": 7
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How to prove $\cos \frac{2\pi }{5}=\frac{-1+\sqrt{5}}{4}$? I would like to find the apothem of a regular pentagon. It follows from
$$\cos \dfrac{2\pi }{5}=\dfrac{-1+\sqrt{5}}{4}.$$
But how can this be proved (geometrically or trigonometrically)?
| Notice that $$\sin 3\theta=3\sin \theta-4\sin^3 \theta$$ Now substitute $\theta=\frac{\pi}{10}$ in the above equation, we get
$$\sin \frac{3\pi}{10}=3\sin \frac{\pi}{10}-4\sin^3 \frac{\pi}{10}$$ $$\implies \cos\left( \frac{\pi}{2}-\frac{3\pi}{10}\right)=3\sin \frac{\pi}{10}-4\sin^3 \frac{\pi}{10}$$ $$\implies \cos\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/7695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "34",
"answer_count": 11,
"answer_id": 1
} |
The Basel problem As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem)
$$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$$
However, Euler was Euler and he gave other proofs.
I believe many of you know some nice proofs of this, can you please share it w... | Following proof rely on this integral identity :
$$\int_{a}^{1}\frac{\arccos x}{\sqrt{x^2-a^2}}\mathrm{d}x=-\frac{\pi}{2}\ln a\qquad ;\,a\in(0,1]$$
We will prove it later on. Now, let's make a power series :
$$\zeta(2)=\sum_{n=1}^{\infty}\frac{1}{n^2}=\int_0^1\frac{1}{x}\sum_{n=1}^{\infty}\frac{x^n}{n}\,\mathrm{d}x=-\i... | {
"language": "en",
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"source": "stackexchange",
"question_score": "814",
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Proving: $\cos A \cdot \cos 2A \cdot \cos 2^{2}A \cdot \cos 2^{3}A ... \cos 2^{n-1}A = \frac { \sin 2^n A}{ 2^n \sin A } $ $$\cos A \cdot \cos 2A \cdot \cos 2^{2}A \cdot \cos 2^{3}A ... \cos 2^{n-1}A = \frac { \sin 2^n A}{ 2^n \sin A } $$
I am very much inquisitive to see how this trigonometrical identity can be pr... | You can prove it by induction since
$\sin t = 2 \cos \dfrac{t}{2}\sin \dfrac{t}{2}$
$\sin t = 2^2 \cos \dfrac{t}{2}\cos \dfrac{t}{4}\sin \dfrac{t}{4}$
$\sin t = {2^3}\cos \dfrac{t}{2}\cos \dfrac{t}{4}\cos \dfrac{t}{8}\sin \dfrac{t}{8}$
So we conjecture:
$$\sin t = 2^n\sin\dfrac{t}{2^n} \prod_{k=1}^{n} \cos\dfrac{t}{2^k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/8439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Find the value of $\displaystyle\sqrt{3} \cdot \cot (20^{\circ}) - 4 \cdot \cos (20^{\circ}) $ How to find the value of
$$\displaystyle\sqrt{3} \cdot \cot (20^{\circ}) - 4 \cdot \cos (20^{\circ})$$
manually ?
| $$\displaystyle \sin(60^{\circ}-20^{\circ}) = \sin 40^{\circ} = 2 \sin 20^{\circ} \cos 20^{\circ}$$
$$\displaystyle \frac{\sqrt{3}}{2} \cos 20^{\circ} - \frac{1}{2} \sin 20^{\circ} = 2 \sin 20^{\circ} \cos 20^{\circ}$$
$$\displaystyle \frac{\sqrt{3}}{2} \cos 20^{\circ} - 2 \sin 20^{\circ} \cos 20^{\circ} = \frac{1}... | {
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"source": "stackexchange",
"question_score": "5",
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How to solve this differential equation? $y''-\frac{y'}{x}=4x^{2}y$ How to solve $y''-\displaystyle\frac{y'}{x}=4x^{2}y$ ?
I know that the solution of this equation is: $y = e^{x^{2}}$, but I cannot resolve.
First I thought that $z=y'$ could be, but was not.
| Note that we have $xy'' - y' = 4x^3y$. The LHS in some sense is dimensional consistent and looks something similar to a quotient rule provided we divide by $x^2$. So dividing by $x^2$ and doing algebraic manipulations we get $(\frac{1}{x}y')' = 4xy$. Now this looks familiar to some extent.
Rewriting, we get $(\frac{y'}... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to compute the formula $\sum \limits_{r=1}^d r \cdot 2^r$? Given $$1\cdot 2^1 + 2\cdot 2^2 + 3\cdot 2^3 + 4\cdot 2^4 + \cdots + d \cdot 2^d = \sum_{r=1}^d r \cdot 2^r,$$
how can we infer to the following solution? $$2 (d-1) \cdot 2^d + 2. $$
Thank you
| As noted above, observe that
\begin{eqnarray}
\sum_{r = 1}^{d} x^{r} = \frac{x(x^{d} - 1)}{x - 1}.
\end{eqnarray}
Differentiating both sides and multiplying by $x$, we find
\begin{eqnarray}
\sum_{r = 1}^{d} r x^{r} = \frac{dx^{d + 2} - x^{d+1}(d+1) + x}{(x - 1)^{2}}.
\end{eqnarray}
Substituting $x = 2$,
\begin{eqnarra... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 9,
"answer_id": 6
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Minimum and maximum There are five real numbers $a,b,c,d,e$ such that
$$a + b + c + d + e = 7$$$$a^2+b^2+c^2+d^2+e^2 = 10$$
How can we find the maximum and minimum possible values of any one of the numbers ?
Source
| Maximum value that can be taken by one of the variables is $\frac{9}{5}$ and the minimum value is $1$.
This can be argued based on symmetry as follows (and I hope I have not overlooked anything in my symmetry argument).
Say we want to find the extreme values of $a$. By symmetry when $a$ reaches the extremum the remaini... | {
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How to prove a formula for the sum of powers of $2$ by induction? How do I prove this by induction?
Prove that for every natural number n, $ 2^0 + 2^1 + ... + 2^n = 2^{n+1}-1$
Here is my attempt.
Base Case: let $ n = 0$ Then, $2^{0+1} - 1 = 1$ Which is true.
Inductive Step to prove is: $ 2^{n+1} = 2^{n+2} - 1$
Our h... | let $S = 2^0 + 2^1 + 2^2 + .... + 2^{n-1}$
so $2S = 2^1 + 2^2 + 2^3 + 2^n$
then $2S - S = S = (2^1 - 2^0) + (2^2 - 2^1) + ... + (2^{n-1} - 2^{n-1}) + 2^n = 2^n - 1$
and we got $2^0 + 2^1 + 2^2 + .... + 2^{n-1} = 2^n - 1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/22599",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "32",
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Probability of 3 of a kind with 7 dice Similar questions:
Chance of 7 of a kind with 10 dice
Probability of getting exactly $k$ of a kind in $n$ rolls of $m$-sided dice, where $k\leq n/2$
Probability was never my thing, so please bear with me.
I've reviewed the threads above to the best of my ability, but I still wonde... | You are on the right track with $6^7 = 279936$ as the denominator. To find how many cases have three but no more matching, I would start by looking at the four partitions of 7 into up to 6 parts where the largest is 3: 3+3+1, 3+2+2, 3+2+1+1, 3+1+1+1+1. You can then work out each systematically, taking account both the... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Identity for $\sum\limits_{j = a}^{N} \binom{N}{j} \binom{j}{a} d^{-j}$? I have run across the following multinomial series:
$$ \sum_{j = a}^{N}
\binom{N}{j} \binom{j}{a} d^{-j}
$$
Here, $d>1$. This seems like a formula which has either a well-known identity, or which has no further closed form or simplif... | Suppose we seek to evaluate
$$\sum_{j=a}^N {N\choose j}{j\choose a} d^{-j}.$$
Start from
$${j\choose a}
= \frac{1}{2\pi i}
\int_{|z|=\epsilon} \frac{1}{z^{a+1}} (1+z)^j \; dz.$$
Note that with $j$ and $a$ integers when $j\lt a$ then $j-a \lt 0$ and
$j-(a+1) \lt -1$ so we may extend the sum to start at $j=0$ because... | {
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"question_score": "6",
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Prove that $x^{2} \equiv 1 \pmod{2^k}$ has exactly four incongruent solutions
Prove that $x^{2} \equiv 1 \pmod{2^k}$ has exactly four incongruent solutions.
My attempt:
We have,
$x^2 - 1 = (x - 1) \times (x + 1)$, then
$(x - 1)(x + 1) \equiv 0 \pmod{2^k}$
which implies,
$2^k|(x - 1)$ or $2^k|(x + 1) \implies x \equ... | Note that $(x - 1)(x + 1) \equiv 0 \mod{2^k}$ will imply that $x$ must be an odd integer. So we may assume that $x = 2m + 1$. Putting value of $x$ in the equation we have $4m(m+1) \equiv 0 \mod{2^k}$. This means that $2^{k-2}$ divides $m(m+1)$ if I assume $k>2$. Note that if $k \leq 2$ then there is no condition on $m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/25128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
"answer_id": 2
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Proving $2 ( \cos \frac{4\pi}{19} + \cos \frac{6\pi}{19}+\cos \frac{10\pi}{19} )$ is a root of$ \sqrt{ 4+ \sqrt{ 4 + \sqrt{ 4-x}}}=x$ How can one show that the number
$2 \left( \cos \frac{4\pi}{19} + \cos \frac{6\pi}{19}+\cos \frac{10\pi}{19} \right)$
is a root of the equation
$\sqrt{ 4+ \sqrt{ 4 + \sqrt{ 4-x}}}=x$?
| Let $$f(x) = 2(\cos \frac{4\pi x}{19} + \cos \frac{6\pi x}{19} + \cos \frac{10\pi x}{19})$$
Then we claim that the following are true:
$$f(1) + f(4)^2 = 4$$
$$f(4) + f(16)^2 = 4$$
$$f(16) + f(64)^2 = 4$$
Noting that $\displaystyle f(4) \lt 0$, $\displaystyle f(16) \lt 0$ and $\displaystyle f(64) = f(1) \gt 0$, we can s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
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If $x^2 \equiv a \pmod{p}$, can we use Jacobi symbol to determine the equation has solution? To determine a quadratic congruence equation has any solution, we have to evaluate
$$\bigg( \dfrac{a}{p} \bigg)$$, So can we apply the algorithm of Jacobi symbol to evaluate this?
If yes, what's the easiest way to evaluate thi... | It looks to me like what you have done is fine. Did you check to see whether 4177 is prime? If it isn't, then Jacobi symbol 1 doesn't guarantee quadratic congruence solvable, e.g., $x^2\equiv2\pmod{15}$ has no solution even though the symbol is 1.
| {
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"url": "https://math.stackexchange.com/questions/33271",
"timestamp": "2023-03-29T00:00:00",
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Writing a percent as a decimal and a fraction I am having a problem understanding some manipulations with recurring decimals. The exercise is
Write each of the following as a
decimal and a fraction:
(iii) $66\frac{2}{3}$%
(iv) $16\frac{2}{3}$%
For (iii), I write a decimal $66\frac{2}{3}\% = 66.\bar{6}\% = 0.66\bar{... | Slightly informally, you just multiply to make the repeating decimals subtract out.
$\begin{align} 10x &=6.\bar{6} \\x &=0.\bar{6} \\9x &=6 \\ x&=\frac{6}{9} \end{align}$
The case for $0.1\bar{6}$ is similar:
$\begin{align} 100x &=16.\bar{6} \\ 10x &=1.\bar{6} \\ 90x &=15 \\ x&=\frac{15}{90} \end{align}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/35631",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
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Formal Power Series Say I differentiate this twice:
$$\dfrac{1}{1+3x} = 1 - 3x + 9x^2 -\cdots+ (-3)^n x^n+\cdots $$
I got
$$\dfrac{18}{(1+3x)^3} = 18 - 162x + \cdots + n\cdot(n-1)(-3)^nx^{n-2}+\cdots$$
If I wanted to get $$\dfrac{1}{(1+3x)^3} = \cdots$$do I just move the 18 over ? Would that work?
$$\frac1{(1+3x)^3}... | Yes. Note that you can write your series as
$$1/(1+3x)^3 = 1 - 9x + \cdots + \dfrac{n(n-1)}{3\cdot3\cdot2}(-3)^nx^{n-2}+\cdots $$
$$1/(1+3x)^3 = 1 - 9x + \cdots + \dfrac{n(n-1)}{ 2}(-3x)^{n-2}+\cdots $$
$$1/(1+3x)^3 = 1 - 9x + \cdots +{n \choose2}(-3x)^{n-2}+\cdots $$
or changing the index
$$1/(1+3x)^3 = 1 - 9x + \cd... | {
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"timestamp": "2023-03-29T00:00:00",
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periodicity of function If $f(x+1) + f(x-1) = \sqrt3f(x)$, then what is the period of $f(x)$?
| $$ f(x+1)+f(x-1) = \sqrt{3} f(x) $$
$$ \sqrt{3}f(x+1) + \sqrt{3}f(x-1)= 3f(x) $$
$$ f(x)+f(x+2)+f(x-2)+f(x)= 3 f(x)$$
$$ f(x+2)+f(x-2)=f(x) $$
$$ f(x+4)+f(x) = f(x+2) $$
Adding last two equations give ,
$$ f(x+4)+f(x-2)= 0$$
$$ f(x+10)+f(x+4)= 0 \implies f(x+10) = f(x-2) \implies f(x+12)=f(x) $$
Thus period = 12... | {
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"timestamp": "2023-03-29T00:00:00",
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Trig identities I need to perform the indicated operation and simplify $(1+\sin t)^{2} + \cos^{2} t$
The book is telling me that it turns into $1 + 2\sin^2t + \cos^2t$, how is is possible? Basic math tells me that 2(3) is equal to six and that $3^2 = 9$ so there is no way that $\sin^2$ can be turned into $2\sin$
| I don't understand what your book is suggesting.
If you first expand the squared binomial, remembering that $(a+b)^2 = a^2 + 2ab + b^2$, we have
$$(1+\sin t)^2 = 1^2 + 2\times 1 \times \sin t + \sin^2 t = 1 + 2\sin t + \sin^2 t.$$
Then, use the fact that $\sin^2 t + \cos^2 t = 1$. So we have:
$$\begin{align*}
(1+\sin... | {
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} |
Seriously: What is the inverse of this 2-by-2 matrix? Okay, I have a hangover and it must be a stupid error, but I just don't get it:
The inverse of a 2-by-2 matrix $A=\left( \matrix{a & b \\ c& d} \right )$ is $\frac{1}{det A}\left( \matrix{d & -b \\ -c& a} \right ) = \frac{1}{ad-bc}\left( \matrix{d & -b \\ -c& a} \r... | You just made a mistake in "... and hence $G^{-1}=$...". Your $\det G$ is correct.
In your problem $\frac{1}{ad-bc}\left( \matrix{d & -b \\ -c& a} \right )=(1+x^2+y^2)\left( \matrix{d & -b \\ -c& a} \right )$. You should know what $a,b,c,d$ are now.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/49216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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A hyperbola as a constant difference of distances I understand that a hyperbola can be defined as the locus of all points on a plane such that the absolute value of the difference between the distance to the foci is $2a$, the distance between the two vertices.
In the simple case of a horizontal hyperbola centred on the... | We will use a little trick to avoid work. We want to have
$$\sqrt{(x+c)^2+y^2} -\sqrt{(x-c)^2+y^2}=\pm 2a.\qquad\text{(Equation 1)}$$
Rationalize the numerator, by multiplying "top" and "bottom" by
$\sqrt{(x+c)^2+y^2} +\sqrt{(x-c)^2+y^2}.$
After the (not very dense) smoke clears, we get
$$\frac{4xc}{\sqrt{(x+c)^2+y^2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/49517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Find $\sin \theta$ and $\cos \theta$ given $\tan 2\theta$ Can you guys help with verifying my work for this problem. My answers don't match the given answers.
Given $\tan 2\theta = -\dfrac{-24}{7}$, where $\theta$ is an acute angle, find $\sin \theta$ and $\cos \theta$
I used the identity, $\tan 2\theta = \dfrac{2\t... | Given $ \tan(2\theta)= -\frac{24}{7}$ From the relation between $\sin(\theta)$, $\cos(\theta)$ and $\tan(\theta)$, we get $$ \frac{\sin(2\theta)}{\cos(2\theta)}= -\frac{24}{7} \implies \sin(2\theta)= -\frac{24}{7} \cos(2\theta)$$and $$ \sin(2\theta)^2 + \cos(2\theta)^2=1$$ $$\cos(2\theta) = \pm \frac{7}{25} \implies 2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/49569",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Trigonometric equality: $\frac{1 + \sin A - \cos A}{1 + \sin A + \cos A} = \tan \frac{A}{2}$ Can you guys give me a hint on how to proceed with proving this trigonometric equality? I have a feeling I need to use the half angle identity for $\tan \frac{\theta}{2}$. The stuff I have tried so far(multiplying numerator and... | $\textbf{Method 1.}$ You have
\begin{align*}
\frac{1+ \sin{A} - \cos{A}}{1+\sin{A} + \cos{A}} &= \frac{ \cos^{2}\frac{A}{2} + \sin^{2}\frac{A}{2} + 2\cdot \sin\frac{A}{2}\cdot\cos\frac{A}{2} - \Bigl(\cos^{2}\frac{A}{2} - \sin^{2}\frac{A}{2}\Bigr)}{\cos^{2}\frac{A}{2} + \sin^{2}\frac{A}{2} + 2\cdot \sin\frac{A}{2}\cdot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/50093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 8,
"answer_id": 5
} |
Analysis of Algorithms: Solving Recursion equations-$T(n)=3T(n-2)+9$ I need your help with solving this recursion equation: $T(n)=3T(n-2)+9$
with the initial condition: $T(1)=T(2)=1.$
I need to find $T(n),$ the complexity of the algorithm which works that way.
I tried to do that with induction.
Edit: The answer should ... | Induction is a way to prove things; it's not generally helpful in solving computational problems, except perhaps in proving that a guessed solution is correct.
Note that the values of $T(n)$ for odd $n$ are completely independent of the values for even $n$, so you can look at it as essentially two separate instances of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/50153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
A fun Pascal-like triangle Inspired by Pascal, I put on some shackles and a thorny belt. Inspiration came pouring in, and I thought of the following triangle:
$$
\begin{array}{rcccccccccc}
& & & & & 1\\\
& & & & 1 & & 1\\\
& & & 1 & & \frac{1}{2} & & 1\\\
& & 1 & & \fra... | The sequence $C_k$ is not monotonic as @JohnM originally claimed. Here's a quick but not so elegant proof of convergence of $C_k$. As a side effect, we'll also know that the sequence goes alternately above and below its limit, namely $1/\sqrt{2}$.
Solving for $C_{k+1}$ in terms of $C_k$, we get $C_{k+1} = \frac{\sqrt{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/53991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "32",
"answer_count": 1,
"answer_id": 0
} |
Is this Batman equation for real? HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real?
Batman Equation in text form:
\begin{align}
&\left(\left(\frac x7\right)^2\sqrt{\frac{||x|-3|}{|x|-3}}+\left(\frac y3\right)^2\sqrt{\frac{\left|y+\frac{3\sqrt{33}}7\right|}{y+\frac{3\sqrt{... | Since people (not from this site, but still...) keep bugging me, and I am unable to edit my previous answer, here's Mathematica code for plotting this monster:
Plot[{With[{w = 3 Sqrt[1 - (x/7)^2],
l = 6/7 Sqrt[10] + (3 + x)/2 - 3/7 Sqrt[10] Sqrt[4 - (x + 1)^2],
h = (3 (Abs[x - 1/2] + Abs[x + 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/54506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "466",
"answer_count": 10,
"answer_id": 2
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Inequality involving sides of a triangle How can one show that for triangles of sides $a,b,c$ that
$$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} < 2$$
My proof is long winded, which is why I am posting the problem here.
Step 1: let $a=x+y$, $b=y+z$, $c=x+z$, and let $x+y+z=1$ to get
$\frac{1-x}{1+x}+\frac{1-y}{1+y}+\f... | $\displaystyle\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} <
\frac{a+a}{b+c+a} + \frac{b+b}{c+a+b} + \frac{c+c}{a+b+c} = 2$
since $a<b+c$ etc.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/54627",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
How to prove the following identities?
Prove:
\begin{align}
\tan(A) + \cot(A) & = 2 \text{cosec}(2A)\\
\tan(45^{\circ}+A^{\circ}) - \tan(45^{\circ}-A^{\circ}) & = 2 \tan(2A^{\circ})\\
\text{cosec}(2A) + \cot(2A) & = \cot(A)
\end{align}
I have got all the formulas that I need but I just couldn't solve these. ... | EDITED. In general there are two different techniques we can use to prove a trigonometric identity $A=B$. One is to transform one side into the other:
$$A=A_1=A_2=\dots =A_n=B.$$
The other is to look at the identity $A=B$ as a whole and convert it into an equivalent one and repeat the process until one known id... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/56122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
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Elementary central binomial coefficient estimates
*
*How to prove that $\quad\displaystyle\frac{4^{n}}{\sqrt{4n}}<\binom{2n}{n}<\frac{4^{n}}{\sqrt{3n+1}}\quad$ for all $n$ > 1 ?
*Does anyone know any better elementary estimates?
Attempt. We have
$$\frac1{2^n}\binom{2n}{n}=\prod_{k=0}^{n-1}\frac{2n-k}{2(n-k)}... | For $n\ge0$, we have (by cross-multiplication)
$$
\begin{align}
\left(\frac{n+\frac12}{n+1}\right)^2
&=\frac{n^2+n+\frac14}{n^2+2n+1}\\
&\le\frac{n+\frac13}{n+\frac43}\tag{1}
\end{align}
$$
Therefore,
$$
\begin{align}
\frac{\binom{2n+2}{n+1}}{\binom{2n}{n}}
&=4\frac{n+\frac12}{n+1}\\
&\le4\sqrt{\frac{n+\frac13}{n+\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/58560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "39",
"answer_count": 7,
"answer_id": 2
} |
Integral question: $\int \frac{x^{5}+x+3}{x{^3}-5x^{2}}\mathrm dx$ i have integral $\int \frac{x^{5}+x+3}{x{^3}-5x^{2}}\mathrm dx$,
so first step is to divide polynoms, and i get:
$\frac{x^{5}+x+3}{x{^3}-5x^{2}}$ = $x^{2} + 5x + 25$ with remainder $125x^2 + x + 3$
is it corrent to divide this polynomial division into... | Yes, it is all correct.
$$
x^5 + x^2 + 3 = 125 x^2 + x + 3 + (x^3-5 x^2)(x^2 + 5 x + 25)
$$
Hence
$$
\begin{eqnarray}
\frac{x^5 + x + 3 }{x^3-5 x^2} &=& x^2 + 5 x + 25 + \frac{125 x^2 +x+ 3}{x^3-5 x} \\ &=&
x^2 + 5 x + 25 -\frac{3}{5 \, x^2} - \frac{8}{25 x} + \frac{3133}{25} \frac{x - 5}{x^2-5} \end{e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/62065",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Simplifying the expression $\sqrt[4]{\frac{g^3h^4}{4r^{14}}}$? How would I simplify this expression?
$$\sqrt[4]{\frac{g^3h^4}{4r^{14}}}\ ?$$
I did this
$$\begin{align*}
\sqrt[4]{g^3h^3h^4}\\
h\sqrt[4]{g^3h^3}\\
\sqrt[4]{4r^{14}}\\
\sqrt[4]{2r^2r^{12}}\\
r^3\sqrt[4]{2r^2}\\
\end{align*}$$
But I am stuck?
Yes that ... | Why did $h^4$ become $h^3h^4$? Why did $4$ become $2$?
In general, remember that for $a$ and $b$ positive, $\sqrt[4]{ab} = \sqrt[4]{a}\sqrt[4]{b}$, and that $(a^r)^{s} = a^{rs}$. Added: If $g$, $h$, and $r$ are positive, then you can rewrite what you have as:
$$\sqrt[4]{\frac{g^3h^4}{4r^{14}}} = \left( g^3\times h^4 \t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/62962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Evaluate $\int \sqrt{1+x^{\frac{3}{2}}} \operatorname d x$ I can't figure out how to integrate
$$\int \sqrt{1+x^{\frac{3}{2}}} \operatorname d x$$
I've tried substitution by letting $u = x^3$, but it didn't go anywhere. I also tried to integrate using a trigonometric substitution, but that also got me nowhere. Then I t... | Let $I = \int \sqrt{1 + x^{3/2}} \, \mathrm{d} x$. Integrating by parts:
$$
\begin{eqnarray}
I &=& x \sqrt{1 + x^{3/2}} - \int x \frac{3}{4} \, \frac{\sqrt{x}}{\sqrt{1+x^{3/2}}} \, \mathrm{d} x = x \sqrt{1 + x^{3/2}} - \frac{3}{4} \int \sqrt{1+x^{3/2}} \, \mathrm{d} x + \frac{3}{4} \int \frac{\mathrm{d} x}{\sqr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/65991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 2,
"answer_id": 0
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How to prove by induction that $a^{2^{k-2}} \equiv 1\pmod {2^k}$ for odd $a$? Let $a \in $ Z be odd. How to prove by induction that $a^{2^{k-2}} \equiv 1\pmod {2^k}$ for all $k \geq 3$.
My attempt: let $k=3$
$$\begin{align*}
a^{2^{k-2}}-1&= a^{2^{3-2}}-1=a^{2}-1\\
& \stackrel{\text{a is odd}}{=} (2l+1)^2-1=4l^2+4l... | For the case $k=3$, you cannot choose $l$. You must argue that $4l(l+1)$ is a multiple of $8$, which can be done by arguing that $l(l+1)$ must be even.
Your induction step seems ok, but it could be written more simply as
$$
a^{2^{n+1-2}}-1 = a^{2^{n-2}\cdot 2}-1=(a^{2^{n-2}}-1)(a^{2^{n-2}}+1)
$$
and use that $a^{2^{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/66043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Help with volume integration I need help solving this integral ($\hat{z}$ denotes the polar axis):$$\int_V\dfrac{\vec{r}\cdot(\vec{r}-c\hat{z})}{|\vec{r}|^3|\vec{r}-c\hat{z}|^3} dV$$ Where $V$ denotes all space.
Attempt: $$2\pi \int_0^\infty \int_0^{\pi}\dfrac{\vec{r}\cdot(\vec{r}-c\hat{z})}{|\vec{r}|^3|\vec{r}-c\hat{z... | This answers the original version of the question:
Use $ \vert \vec{r} - c \hat{z} \vert^2 = r^2 + c^2 - 2 r c \cos \theta$. Now integration with respect to $\theta$ can be carried out, change $t = \cos\theta$.
$$
\begin{eqnarray}
2 \pi \int_0^ \pi \frac{r(1-\cos \theta)}{ \left( r^2 + c^2 - 2 \, c \cdot r \cdot \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/66139",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
evaluating $ \int_0^{\sqrt3} \arcsin(\frac{2t}{1+t^2}) \,dt$ $$\begin{align*}
\int \arcsin\left(\frac{2t}{1+t^2}\right)\,dt&=t\arcsin\left(\frac{2t}{1+t^2}\right)+\int\frac{2t}{1+t^2}\,dt\\
&=t\arcsin\left(\frac{2t}{1+t^2}\right) + \ln(1+t^2)+C
\end{align*}$$
So
$$ \int\nolimits_0^{\sqrt3} \arcsin\left(\frac{2t}{1+t^... | The Weierstrass substitution in a slightly different form from that in which I'm accustomed to seeing it will do it.
We have
$$
\begin{align}
t & = \tan\frac x2 \\ \\
\frac{2\;dt}{1+t^2} & = dx \\ \\
\frac{2t}{1+t^2} & = \sin x \\ \\
\frac{1-t^2}{1+t^2} & = \cos x
\end{align}
$$
That's the usual Weierstrass s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/66457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
equation of the tangent to the following curves? example:
$f(x)=x^2 (2,4)$
$= x^2$
at x= 2
$f(x) 2^2=4$
$y=mx+c$
$y=4x+c$
$(2,4)$
$4=4\cdot2+c$
$4-8=c$
$c=-4$
$y=4x-4$
but below question
I don't know how to solve?
$F(x)=x^2-2x+5$ at $x= -1$
the answer is $y=-4x+4$
thanks in advance! :)
| Added: concerning the first part of your question, you should edit it. As a minimum, something like the following.
Example: $f(x)=x^2$. At $x=2$, $f(x)=2^2=4$. The equation of the tangent at $(2,4)$ is
$$y=mx+c,$$
where $m$ is $f'(2)=4$. And so, $y=4x+c$, and $$4=4\cdot 2+c\Leftrightarrow 4-8=c\Leftrightarrow c=-4.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/68441",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
challenging alternating infinite series involving $\ln$ I ran across an infinite series that is allegedly from a Chinese math contest.
Evaluate:
$\displaystyle\sum_{n=2}^{\infty}(-1)^{n}\ln\left(1-\frac{1}{n(n-1)}\right).$
I thought perhaps this telescoped in some fashion. So, I wrote out
$\ln(1/2)-\ln(5/6)+\ln(11/12)... | Use $ \log\left(1 - \frac{1}{n(n-1)}\right) = \int_0^1 \frac{\mathrm{d} t}{n(1-n)+t}$.
Then
$$ \begin{eqnarray}
\sum_{n=2}^\infty \frac{(-1)^n}{n - n^2 + t} &=& \sum_{n=2}^\infty \frac{2 (-1)^n }{\sqrt{4 t+1}} \left(\frac{1}{2 n+\sqrt{4 t+1}-1}-\frac{1}{2 n-\sqrt{4
t+1}-1}\right) \\&=&
\frac{1}{2 \sqrt{4 t+1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/69090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
"answer_id": 0
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How to convert a fraction into ternary? In a ternary system how is $\dfrac{1}{2}=0.\bar1$ ,$\dfrac{1}{3}=0.1=0.0\bar2$??, etc. In general, how does one write the ternary expression for a given fraction?
| I assume you know how to convert integers from base 10 to base 3, so this answer will address fractions between $0$ and $1$.
To say, for example, that $\frac{5}{8} = 0.625$ means that $\frac{5}{8} = 6 \frac{1}{10} + 2 \frac{1}{100} + 5 \frac{1}{1000}$. So to convert a fraction $\frac{a}{b}$ to ternary means we want to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/70382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 2,
"answer_id": 0
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Why do definitions of distinct conic sections produce a single equation? I understand how to get from the definitions of a hyperbola — as the set of all points on a plane such that the absolute value of the difference between the distances to two foci at $(-c,0)$ and $(c,0)$ is constant, $2a$ — and an ellipse — as the ... | Notice that E and H have two constants, A has three. So A forms a superset of E and H and you should see graph of A to study how E and H are fitted together to form A in a certain composite geometrical pattern/relationship.
Because a certain manipulation on A can bring it into form of E or H, we can see that a three pa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/70732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Proof of dividing fractions $\frac{a/b}{c/d}=\frac{ad}{bc}$ For dividing two fractional expressions, how does the division sign turns into multiplication? Is there a step by step proof which proves
$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}=\frac{ad}{bc}?$$
| The fraction $x = \frac{a}{b}$ is the solution to $b \cdot x = a$. Similarly $y = \frac{c}{d}$ solves $d \cdot y = c$. Now, multiply the left-hand-side with $b \cdot x$ and the right hand-size with $a$ which preserves the equality since $a = b \cdot x$. We get
$$
\begin{eqnarray}
(b \cdot x) \cdot ( d \cdot y) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/71157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 1
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How do I prove that $\cos{\frac{2\pi}{7}}\notin\mathbb{Q}$? How do I prove that $\cos{\frac{2\pi}{7}}\notin\mathbb{Q}$?
Should I use some geometrical approach or apagoge?
| Since Chebyshev polynomials have the property $T_n\left( \cos \theta \right) = \cos(n \theta)$,
we can use this property with $n=7$ and $\theta = \frac{2 \pi}{7}$.
Let $z = \cos\left( \frac{2\pi}{7} \right)$. Then $T_7(z) = 1$
Let $p(x) = T_7(x)-1 = 64 x^7 - 112 x^5 +56 x^3 - 7x - 1 = (x-1)( 8 x^3 + 4 x^2 - 4 x -1 )^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/71698",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 5,
"answer_id": 4
} |
Partial Fractions of form $\frac{1}{(ax+b)(cx+d)^2}$ When asked to convert something like $\frac{1}{(ax+b)(cx+d)}$ to partial fractions, I can say
$$\frac{1}{(ax+b)(cx+d)} = \frac{A}{ax+b} + \frac{B}{cx+d}$$
Then why can't I split $(cx+d)^2$ into $(cx+d)(cx+d)$ then do
$$\frac{1}{(ax+b)(cx+d)^2} = \frac{A}{ax+b} + \fr... | Because
$$\frac{B}{cx+d}+ \frac{C}{cx+d} = \frac{B+C}{cx+d}\ne\frac{1}{(cx+d)^2}$$ for any $B$ and $C$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/72281",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
$1\cdot 3 + 2\cdot 4 + 3\cdot 5 + \cdots + n(n+2) = n(n+1)(2n+7)/6$ by mathematical induction I am doing mathematical induction. I am stuck with the question below. The left hand side is not getting equal to the right hand side.
Please guide me how to do it further.
$1\cdot 3 + 2\cdot 4 + 3\cdot 5 + \cdots + n(n+2) = ... | As I understand it you are trying to show that for $n\geq 1$:
$$
1\cdot 3+2\cdot 4+\cdots +n(n+2)=\frac{1}{6}n(n+1)(2n+7).
$$
You first showed it for $n=1$:
$$
3=1\cdot 3=\frac{1}{6}(1)(2)(9)=3
$$
Now we assume that the formula holds for some $n\geq 1$ and we have the statement for $n+1$
$$
3\cdot 1+\cdots +n(n+1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/72660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Taylor expansion of $ \arccos(\frac{1}{\sqrt{2}}+x)$, $ x\rightarrow0$ What is the method to calculate the Taylor expansion of $ \arccos(\frac{1}{\sqrt{2}}+x)$, $ x\rightarrow0$ ?
| The formula for the cosine of a difference yields
$$
\begin{align}
\cos(\pi/4-y)
&= \frac{1}{\sqrt{2}}\cos(y)+\frac{1}{\sqrt{2}}\sin(y)\\
&=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}(\sin(y)+\cos(y)-1)\\
&=\frac{1}{\sqrt{2}}+x\tag{1}
\end{align}
$$
Noting that $x=\frac{1}{\sqrt{2}}(\sin(y)+\cos(y)-1)$, it is easy to ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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Evaluating an expression using snake oil I have to evaluate this expression: $\sum_k\binom{n}{k}\binom{2k}{k}(-2)^{-k}$,
(In the original question we had $\sum_k\binom{n}{k}\binom{2k}{k}(-2)^{k}$)
this is what I have done:
$$\begin{aligned}
\sum_n\sum_k\binom{n}{k}\binom{2k}{k}(-2)^{-k}x^n
& = \sum_k\sum_n\binom{n}{k... | Added: This post answers the original question, which is to compute $\sum_{k=0}^\infty \binom{n}{k} \binom{2k}{k} (-2)^k$.
Assuming the typo is corrected you correctly arrived at the result:
$$
\sum_{n=0}^\infty c_n x^n = \frac{1}{\sqrt{(1-x)(1+7 x)}}
$$
Where $c_n = \sum_{k=0}^\infty \binom{n}{k} \binom{2k}{k} (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/78084",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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The limit of infinite product $\prod_{k=1}^{\infty}(1-\frac{1}{2^k})$ is not 0? Is there any elementary proof that the limit of infinite product $\prod_{k=1}^{\infty}(1-\frac{1}{2^k})$ is not 0?
| This is similar to the beginning of Alex's answer, however it proceeds differently.
We express the log of the product as a sum of logarithms
$$\small L=-\log(\prod_{k=1}^\infty (1-{1 \over 2^k})) = -\sum_{k=1}^\infty \log(1-{1 \over 2^k})$$
Next we expand each of the logs according to its power series and write this a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/78689",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 2
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Finding pairs of integers such that $x^2+3y$ and $y^2+3x$ are both perfect squares Can we find pairs $(x,y)$ of positive integers such that $x^2+3y$ and $y^2+3x$ are simultaneously perfect squares? Thanks a lot in advance. My progress is minimal.
| $x^2+3y=a^2$
$y^2+3x=b^2$
$a,b \in \mathbb{N}$
assume wlog $x\ge y$
$a^2> x^2$
$a > x$ but since $x\ge y$ we have $x+2>a$
so $a=x+1 $
$3y=2x+1$
$x=3k+1 ,y=2k+1$
$4k^2+4k+1+9k+3=b^2$
solving for k we get
$16b^2+105=t^2$ is a perfect square
$b=1,2,4,13$
so $b=2,4,13$ and $x=y=1$ or $x=16 $, $y=11 $ and if $y\ge x$ we ca... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
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Is this proof about the form $2^n \pm a$ correct? I want to prove following statement :
For prime numbers $p$ greater than $3$, it is true that:
$a)$ if $p=2^n-a$ and $a=6k+1$, then $n$ is an odd number.
$b)$ if $p=2^n+a$ and $a=6k-1$, then $n$ is an odd number.
$c)$ if $p=2^n-a$ and $a=6k-1$, then $n$ is an even num... | Yes, the proofs are fine. You did not do part (d), but it requires nothing new, so it is quite reasonable to omit it. In the formal statement of the lemma, the conditions on the parameters should have been made explicit.
You chose to avoid congruence notation. I would probably instead observe first that $2^n \equiv 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/81900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Proving that a polynomial is positive A Finnish mathematics competition asked to prove that for all $x$ we have $x^8-x^7+2x^6-2x^5+3x^4-3x^3+4x^2-4x+\frac{5}{2}\geq 0$ for all real $x$. I heard that it follows from Hilbert's problem that one can prove this by writing the polynomial as sum of squares. How can I find suc... | I'm not sure if this is what you're looking for, but it works and is quite elementary.
We can take the highest three powers and do this:
$$
x^8-x^7+x^6 = x^6(x^2-x+1) = x^6\left[\left(x-\dfrac{1}{2}\right)^2+\dfrac{3}{4}\right]
= x^6 \left(x-\dfrac{1}{2}\right)^2 + \dfrac{3}{4} x^6,
$$
which gives us the expression
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/83670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
variance of a sum We draws 10 from a finite population of 30, without replacement! 15 members out of 30 have the number 1, 10 members have the number 2 and 5 members have the number 3.
The Expected value is
$$\mathbb E(X)= 1 \cdot \frac12+2 \cdot \frac13+3 \cdot \frac 16 = \frac 53 $$
Variance
$$V(X)= \left( 1 -\frac5... | It is simpler to work it out directly. Since the total of the sample $Z = X_1+\ldots+X_{10}$ does depend on the order of sampling of each individual components, the distribution of $Z$ would be the same if, instead of sampling without replacement, we sampled 10 elements at once.
Suppose a sample contains $k_1$ balls wi... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proving $2,3,1+\sqrt{-5}$ and $1-\sqrt{-5}$ are irreducible in $\mathbb{Z}[\sqrt{-5}]$ Could anyone help me prove that $2,3,1+\sqrt{-5}$ and $1-\sqrt{-5}$ are irreducible in $\mathbb{Z}[\sqrt{-5}]$?
As $6=2*3=(1+\sqrt{-5})(1-\sqrt{-5})$ so $\mathbb{Z}[\sqrt{-5}]$ is not a UFD. Therefore is not a PID or euclidean domain... | Since $\sqrt{-5} \cdot \sqrt{-5}=-5 \in \mathbb{Z}$ holds, we have $\mathbb{Z}[\sqrt{-5}]=\{a+b \sqrt{-5} \mid a, b \in \mathbb{Z}\}.$
We use
$$
(a+b \sqrt{-5})(c+d \sqrt{-5})=a c-5 b d+\left(a d+bc\right) \sqrt{5}
$$
and $|a+b \sqrt{-5}|^2=a^2+5 b^2$.
$1+\sqrt{-5}$ is not a unit: it is valid that
$$
(1+\sqrt{-5})(a+b ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/86383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
"answer_count": 3,
"answer_id": 2
} |
How to find $n$'th term of the sequence $3, 7, 12, 18, 25, \ldots$? $$3, 7, 12, 18, 25, \ldots$$
This sequence appears in my son's math homework. The question is to find the $n$'th term. What is the formula and how do you derive it?
| for the series: 3, 7, 12, 18, 25, …
$3=0+3\\7=3+4\\12=7+5\\18=12+6\\25=18+7$
We can find that: each term equals previous term plus n+2
$t_0=0$
$t_1=t_0+(1+2)=(1+2)$
$t_2=t_1+(2+2)=(1+2)+(2+2)$
$t_3=t_2+(3+2)=(1+2)+(2+2)+(3+2)$
...
$t_n=t_{n-1}+(n+2)$
$=(\color{red}1+\color{green}2)+(\color{red}2+\color{green}2)+(\color... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/88277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
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How to compute the characteristic polynomial of $A$ The matrix associated with $f$ is:
$$ \left(\begin{array}{rrr}
3 & -1 & -1 \\
-1 & 3 & -1 \\
-1 & -1 & 3
\end{array}\right) .
$$
First, I am going to find the characteristic equation of $\det(A- \lambda I)$. Please correct me if I'm wrong.
$$= (3-\... | You may also use Cardano's formula to find the roots $x_1,x_2,x_3$ of a cubic polynomial $ax^3+bx^2+cx+d$.
$$
\begin{align}
p &= 2 b^3-9 a b c+27 a^2 d,\\
q &= \sqrt{p^2-4 \left(b^2-3 a c\right)^3},\\
x_1 &= -\frac{b}{3 a}
-\frac{1}{3 a} \sqrt[3]{\frac{p+q}{2}}
-\frac{1}{3 a} \sqrt[3]{\frac{p-q}{2}},\\
x_2 &= -\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/88324",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
} |
$\int \cos^{-1} x \; dx$; trying to salvage an unsuccessful attempt $$
\begin{align}
\int \cos^{-1} x \; dx &= \int \cos^{-1} x \times 1 \; dx
\end{align}
$$
Then, setting
$$\begin{array}{l l}
u=\cos^{-1} x & v=x \\
u' = -\frac{1}{\sqrt{1-x^2}} & v'=1\\
\end{array}$$
Then by the IBP technique, we have:
$$\begin... | Put $x = \cos{\nu}$. Then you have $dx = -\sin\nu \ d\nu$. So your integral is now $$I =\int \nu \cdot \sin\nu \ d\nu$$
This is easy to evaluate by parts. Take $u = \nu$ and $dv = \sin\nu \ d\nu$. Then you have
\begin{align*}
I &= uv - \int v \ du \\ &= \Bigl[ -\nu \cdot \cos\nu \Bigr] + \int \cos\nu \ d\nu
\end{alig... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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Is this a proper use of induction? ($(n^2+5)n$ is divisible by 6) Just want to get input on my use of induction in this problem:
Question. Use mathematical induction to prove that $(n^2+5)n$ is divisible by $6$ for all integers $n \geqslant 1$.
Proof by mathematical induction.
(1) show base case ($n=1$) is true:
$$
... | The statement that
$3n^2+3n$ is divisible by $6$ for all $n$.
and the statement that
$3(1)^2+3(1)=6$ is divisible by $6$.
are clearly not the same :) You can prove the former statement by showing that $n^2+n=n(n+1)$ is always even (use that either $n$ is even, or $n+1$ is).
By the way, use equals signs ($=$) inste... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/90064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Right triangle where the perimeter = area*k I was doodling on some piece of paper a problem that sprung into my mind. After a few minutes of resultless tries, I advanced to try to solve the problem using computer based means.
The problem stated is
Does a right angle triangle with integer sides such that
$$P = A\c... | If you choose positive integers $a$ and $b$, you can make a right triangle with sides
$a^2-b^2\text{, } 2ab \text{ and } a^2+b^2,$
and all right triangles can be found this way by varying the values of $a$ and $b$.
If we have such a triangle, its area is
$$\frac{1}{2}(a^2-b^2)\times 2ab = ab(a^2-b^2)$$
and its perime... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/90236",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 2
} |
Trigonometric integral I would like to know if there is some way to calculate exactly
$$\int_{0}^{\pi/2} \frac{\cos^3 x}{\sqrt {\sin x}(1+\sin^3 x)}\,dx$$
The numerical value is 1,52069
| From Dilip Sarwate's suggestion on:
$$2\int_0^1\frac{1-y^4}{1+y^6}\mathrm dy = 2\int_0^1\frac{(1-y^2)(1+y^2)}{(1+y^2)(y^4-y^2+1)}\mathrm dy = 2\int_0^1\frac{(1-y^2)}{(y^4-y^2+1)}\mathrm dy \; .$$
The denominator can be split further into
$$2\int_0^1\frac{(1-y^2)}{(y^4-y^2+1)}\mathrm dy = 2\int_0^1\frac{(1-y^2)}{(y^2+\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/94095",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Converting to partial fractions I would like to see a step by step conversion of this fraction to two partial fractions.
$\frac{x^2}{1-x^2}$
| As André said, because the degree of the numerator is greater than or equal to that of the denominator, the first step is to divide out to get polynomial quotient and a rational function remainder whose numerator has smaller degree than its denominator:
$$\frac{x^2}{1-x^2}=-1+\frac1{1-x^2}\;.$$
Now expand the second te... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/98340",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The inequality $b^n - a^n < (b - a)nb^{n-1}$ I'm trying to figure out why $b^n - a^n < (b - a)nb^{n-1}$.
Using just algebra, we can calculate
$ (b - a)(b^{n-1} + b^{n-2}a + \ldots + ba^{n-2} + a^{n-1}) $
$ = (b^n + b^{n-1}a + \ldots + b^{2}a^{n-2} + ba^{n-1}) - (b^{n-1}a + b^{n-2}a^2 + \ldots + ba^{n-1} + a^{n-1}) $
$ ... | This inequality can fail if $a=b$ or $a$ and $b$ have differing signs: e.g. $b=1$, $a=-3$, and $n=3$. So let's assume that $a\not=b$ and $a,b\ge0$. Division yields
$$
\frac{b^n-a^n}{b-a}=\sum_{k=1}^nb^{n-k}a^{k-1}\tag{1}
$$
If $a<b$ then obviously, $\sum\limits_{k=1}^nb^{n-k}a^{k-1}=b^{n-1}\sum\limits_{k=1}^n\left(\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/104027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Trigonometric equality $\cos x + \cos 3x - 1 - \cos 2x = 0$ In my text book I have this equation:
\begin{equation}
\cos x + \cos 3x - 1 - \cos 2x = 0
\end{equation}
I tried to solve it for $x$, but I didn't succeed.
This is what I tried:
\begin{align}
\cos x + \cos 3x - 1 - \cos 2x &= 0 \\
2\cos 2x \cdot \cos x - 1 - \... | You can also use the identity:
$$\cos(A)+\cos(B) = 2 \cos(\frac{A+B}{2}) \cos (\frac{A-B}{2})$$
Thus
\begin{equation}
\begin{split}
\cos x + \cos 3x - 1 - \cos 2x &= \cos x + \cos 3x - \cos(0) - \cos 2x \
&= 2\cos(\frac{x+3x}{2})\cos(\frac{x-3x}{2})- 2\cos(\frac{0+2x}{2})\cos(\frac{0-2x}{2})
\end{split}
\end{equation... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/104350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Simplifying $1 - x + x^2 - x^3 + ... + x^{98} - x^{99}$ to an equivalent expression. I am doing an exercise to see the error when solving this polynomial for $x = 1.00001$ using nested multiplication.
I believe the correct way to achieve this simplification (based on a lecture) is to multiply the polynomial by $\frac{1... | Here's the beginning:
$$\begin{align}
&\ \ \ 1-x+x^2-x^3+\cdots+ x^{98}-x^{99}\\
&= \left(1-x+x^2-x^3+\cdots+ x^{98}-x^{99}\right)\frac{1+x}{1+x}\\
&= \frac{1(1+x)-x(1+x)+x^2(1+x)+\cdots +x^{98}(1+x)-x^{99}(1+x)}{1+x}\\
&=\frac{1+x-x-x^2+x^2+x^3-x^3\cdots +x^{98}+ x^{99}-x^{99} -x^{100}}{1+x}\\
\end{align}$$
Can y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/104889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
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Diophantine equation $x^3+z x^2-z y^2=0$ I'm not familiar with diophantine equations. At most my approaches doesn't give results. I need to solve the following equation
$$
x^3+zx^2-zy^2=0
$$
where $x,y,z\in\mathbb{Z}$
| We have $x^2 (x+z) = z y^2.$ If $x=0$ things simplify, so let us assume that $x,y,z \neq 0$ until such time as we can work those in.
$$ x+z = z \; \frac{y^2}{x^2}.$$
Define a rational number $r = \pm y / x,$ so
$$ x + z = r^2 z. $$ Then
$$ x = (r^2 -1)z, $$ and
$$ y = \pm x = \pm r (r^2 -1)z. $$
ORIGINAL: All ra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/105860",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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System $a+b+c=4$, $a^2+b^2+c^2=8$. find all possible values for $c$. $$a+b+c=4$$$$a^2+b^2+c^2=8$$
I'm not sure if my solution is good, since I don't have answers for this problem. Any directions, comments and/or corrections would be appreciated.
It's obvious that $\{a,b,c\}\in[-\sqrt8,\sqrt8]$. Since two irrational num... | All the points are given by
$$ \left( \frac{4}{3}, \; \frac{4}{3}, \; \frac{4}{3} \; \right) + \frac{\sqrt{12}}{3} \; \left(1, \; -1, \; 0 \; \right) \; \cos t + \frac{2}{3} \; \left(1, \; 1, \; -2 \; \right) \; \sin t $$ where the points with a zero and a pair of 2's occur at $t = \frac{\pi}{2},\frac{7 \pi}{6... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/106530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
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Manipulating complex numbers Express the complex number $(-1-i)^{48}(-3+i\sqrt{3})^{36}$ in the form a + bi?
I got $ (2^{24})(12^{18}) + 0i $?
Did I get it right?
| Note: $\:x\: =\: -1\:-\:i\: \Rightarrow\:x^4 = (2i)^2 =\: -4.\ $ Recall $\:\zeta^3 = 1\:$ for $\:\zeta = (-1 - \sqrt{-3})/2\:$
thus $\:y\: =\: \sqrt{-3} - 3\ \Rightarrow\: y^6 \: =\: \left(\sqrt{-3}\: (1+\sqrt{-3})\right)^6\ =\ (\sqrt{-3}\:(-2\:\zeta))^6 \:=\: - 2^6\cdot 3^3$
Therefore $\:x^{48} y^{36} = (x^8 y^6)^6 =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/109353",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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The sum of the coefficients of $x^3$ in $(1-\frac{x}{2}+\frac{1}{\sqrt x})^8$ I know how to solve such questions when it's like $(x+y)^n$ but I'm not sure about this one:
In $(1-\frac{x}{2}+\frac{1}{\sqrt x})^8$, What's the sum of the
coefficients of $x^3$?
| $$
\begin{align}
\left(\frac1{\sqrt{x}}+1-\frac12x\right)^8
&=\sum_{k=0}^8\binom{8}{k}\left(\frac1{\sqrt{x}}\right)^k\left(1-\frac12x\right)^{8-k}\\
&=\color{#00A000}{\sum_{k=0}^4\binom{8}{2k}\left(\frac1x\right)^k\left(1-\frac12x\right)^{8-2k}}\\
&+\color{#C00000}{\frac1{\sqrt{x}}\sum_{k=0}^3\binom{8}{2k+1}\left(\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/109748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 0
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Possibility to simplify $\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{\pi }{{\sin \pi a}}} $ Is there any way to show that
$$\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{1}{a} + \sum\limits_{k = 1}^\infty {{{\left( { - 1} \righ... | This is a very elegant and quick way to evaluate this sum with complex analysis. Consider
$$g(z) = \pi \csc (\pi z)f(z)$$
$\csc$ has poles at $2 \pi n$ and $2 \pi n + \pi$ for $n \in \mathbb Z$. Assuming $f(z)$ has no poles at any integer, the residue of $g(z)$ at $2\pi n$ is
$$\operatorname*{Res}_{z = 2 n} g(z) = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/110494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "42",
"answer_count": 4,
"answer_id": 3
} |
smallest ODD number n whose sum of divisors (including n) is at least 3n As usual, for a positive integer $n$, let $\sigma(n)$ denote the sum of all positive divisors of $n$ (including 1 and $n$).
What is the smallest ODD number such that $\sigma(n) \ge 3n$?
For comparison, the answers to some related questions are:
Sm... | Nope, the smallest is
$$
n=1018976683725=3^3 \cdot 5^2 \cdot 7^2 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29,
$$
which has $\frac{\sigma(n)}{n}=\frac{34283520}{11350339}\approx 3.02048423399513$.
Just for the record, I found this by using your observation that any candidate must be a multiple of $m:=3 \cdo... | {
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"timestamp": "2023-03-29T00:00:00",
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Showing $ \sum_{n=0}^{\infty} \frac{1}{(3n+1)(3n+2)}=\frac{\pi}{3\sqrt{3}}$ I would like to show that:
$$ \sum_{n=0}^{\infty} \frac{1}{(3n+1)(3n+2)}=\frac{\pi}{3\sqrt{3}} $$
We have:
$$ \sum_{n=0}^{\infty} \frac{1}{(3n+1)(3n+2)}=\sum_{n=0}^{\infty} \frac{1}{3n+1}-\frac{1}{3n+2} $$
I wanted to use the fact that $$\arcta... | An important trick here is that sigma and integral signs can be changed around.
$$\int \sum^b_{n=a} f\left(n,x\right)\, dx = \sum^b_{n=a} \int f\left(n,x\right) \,dx$$
And this is because
$$\int \sum^b_{n=a} f(n,x)\, dx$$
$$\int f\left(a,x\right) + f((a+1),x) + f((a+2),x) + \dots +f\left((b-1),x\right) + f(b,x) $$
$$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/112161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 6,
"answer_id": 4
} |
Finding a limit of a series How would you calculate this limit it just blew me off on my midterms i seem to have calculated the limit correctly but my process is bougus < what my friend said.
$$
\lim_{n\to\infty}\frac{n \sqrt{n} +n}{\sqrt{n^3}+2}
$$
How i calculated the limit:
$$
\lim_{n\to\infty}\frac{n \sqrt{n^2 ... | I didn't get how it appeared $\sqrt{n^2+\frac1n}$ already in the first numerator - but the answer you got is correct. Nevertheless, for such limits with $n\to\infty$ and powers both in numerator and denominator, you should always divide both numerator and denominator by the highest power of $n$. Here the highest power ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/112463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Gaussian curvature of an ellipsoid proportional to fourth power of the distance of the tangent plane from the center? Is it true that the Gaussian curvature of an ellipsoid is proportional to fourth power of the distance of the tangent plane from the center? I can verify that it holds at the places where the major axes... | I did the same thing for the "elliptic hyperboloids," of one sheet or two sheets, see
http://mathworld.wolfram.com/EllipticHyperboloid.html
where they have some pictures and formulas. There is enough detail, it is not necessary to use the parametrizations.
I am going to use a fixed number $$ \delta = \pm 1$$ and surf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/112662",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
What is $\lim_{n\rightarrow \infty} \frac{1}{n^2}(\ln(\frac{2^n}{3^n})+\ln(\frac{5^n}{4^n})+\cdots+\ln(\frac{(3n-1)^n}{(n+2)^n}))$? Per the title of this question, how does one go about calculating $$\lim_{n\rightarrow \infty} \frac{1}{n^2}\left(\ln\left(\frac{2^n}{3^n}\right)+\ln\left(\frac{5^n}{4^n}\right)+\cdots+\ln... | For a mechanical* way to do this:
Hint:
$$\log \frac{(3k-1)}{k+2} = \log 3 + \log (1- \frac{1}{3k}) - \log (1 + \frac{2}{k})$$
and
$$\log (1-x) = -x - \frac{x^2}{2} - \frac{x^3}{3} - \dots $$
for $|x| \lt 1$.
More details:
Using the above hint, the $k^{th}$ term is
$$ n \log \frac{(3k-1)}{k+2} = n \log 3 + \frac{5n}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/113446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Evaluate $\sum_{n=0}^\infty \frac1{n^3+1}$ if it can express in terms of elementary functions. Evaluate $\sum_{n=0}^\infty \frac1{n^3+1}$ if it can express in terms of elementary functions .
You may refer to this and this articles.
| Use $n^3+1 = (n+1)(n^2-n+1) = (n+1)(n-\omega)(n - \bar{\omega})$, where $\omega = \mathrm{e}^{i \pi/3}$. Then
$$
\frac{1}{n^3+1} = \frac{1}{3} \frac{1}{n+1} - \frac{1}{3} \frac{\omega}{n -\omega} - \frac{1}{3} \frac{\bar{\omega}}{n -\bar{\omega}} = \frac{1}{3} \frac{\omega +\bar{\omega}}{n+1} - \frac{1}{3} \frac{\o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/113986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
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