Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
A problem with logarithms If $\log(a+b+c)=\log(a) + \log(b) + \log(c)$, prove that $$\log\left(\frac{2a}{1-a^2} +\frac{2b}{1-b^2} +\frac{2c}{1-c^2}\right) = \log\left(\frac{2a}{1-a^2}\right)+ \log\left(\frac{2b}{1-b^2}\right)+\log\left(\frac{2c}{1-c^2}\right) $$
| Assuming $a,b,c>0$, Let:
$a =\tan\alpha $
$b =\tan\beta $
$c =\tan\gamma$
Inserting it in the given relation of the question, we get:-
$$\log(a+b+c)=\log(a) + \log(b) + \log(c)$$
$$=> a+b+c= abc => \tan\alpha +\tan\beta+\tan\gamma=\tan\alpha\tan\beta\tan\gamma$$
$$=> \frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1264309",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to simplify Boolean Expression $\bar B + \bar C (B + A)$ I trying to figure out how $ \bar B + \bar C (B + A)$ simplifies to
$ \bar B + \bar C$.
| It's usually easier to figure these things out with Karnaugh maps. After playing around, we see that:
\begin{align*}
\overline B + \overline C(B + A)
&= \overline B(1) + \overline C(B + A) \\
&= \overline B(1 + \overline C) + \overline C(B + A) \\
&= \overline B + \overline B ~ \overline C + \overline C(B + A) \\
&= \o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1265685",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Cyclic Equation. Prove that: $\small\frac { a^2(b-c)^3 + b^2(c-a)^3 + c^2(a-b)^3 }{ (a-b)(b-c)(c-a) } = ab + bc + ca$? This is how far I got without using polynomial division:
\begin{align}
\tiny
\frac { a^{ 2 }(b-c)^{ 3 }+b^{ 2 }(c-a)^{ 3 }+c^{ 2 }(a-b)^{ 3 } }{ (a-b)(b-c)(c-a) }
&\tiny=\frac { { a }^{ 2 }\{ { b }^{ 3... | If you don't feel like being clever, brute forcing will also work: the numerator
\begin{align*}
&=a^2(b-c)^3 + b^2(c-a)^3 + c^2(a-b)^3\\
&=a^2[(b-a)+(a-c)]^3+b^2(c-a)^3+c^2(a-b)^3\\
&=a^2(b-a)^3+3a^2(b-a)(a-c)(b-c)+a^2(a-c)^3+b^2(c-a)^3+c^2(a-b)^3\\
&=-(c-a)^3(a-b)(a+b)+(a-b)^3(c-a)(c+a)+3a^2(b-a)(a-c)(b-c)
\end{align*... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1267504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Find all pairs of positive integers $(m,n)$ such that $2^{m+1}+3^{n+1}$ is a perfect square Find all pairs of positive integers $(m,n)$ such that $2^{m+1}+3^{n+1}$ is a perfect square
My attempt so far
Any perfect square is $0,1$ in mod 4, so $n+1$ must be even :
$$2^{m+1}+3^{2r}=k^2$$
Rearranging and factoring
$$2^{m... | All perfect squares are either $0$ or $1 \pmod 3$ so $m+1$ is also even. We now have:
$$2^{2s} + 3^{2r} = k^2$$
Or, equivalently:
$$(2^s)^2 + (3^r)^2 = k^2$$
Wich is a pythagorean triple.
Since $(2^s,3^r) = 1$ then there are integers $p,q$ so that:
$$\begin{cases}2pq = 2^s & \\
p^2 - q^2 = 3^r & \\
p^2 + q^2 = k &
\end... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1269526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Did I do this Continuous Probability Problem Correctly? I'm new to evaluating continuous probability density functions. I'd like someone to check my work, please.
Problem:
Suppose $X$ has density $f(x) = c/x^6$ for $x>1$ and $f(x) = 0$
otherwise, where $c$ is a constant.
*
*Find $c$.
*Find $E(X)$.
*Fi... | Your reasoning and execution are perfectly correct.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1272281",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Simple Lagrange Multiplyers Problem Can anyone please help me with the following:
Find the stationary values of $u=x^2+y^2$ subject to the constraint $t(x,y) = 4x^2 + 5xy + 3y^2 = 9$.
The answer is given as $u = 9$ and $x = \pm 3/\sqrt2$ and $y = \mp 3/\sqrt2$
I do the following:
Let $g(x,y) = 4x^2 + 5xy + 3y^2 - 9 = ... | I haven't worked out a numerical answer by hand:
if I eliminate k from:
$$\begin{align*}
2x + k(8x + 5y) &= 0\\
2y + k(5x + 6y) &= 0\\
\end{align*}$$
I get:
$$\begin{align*}
2y - \frac{2x(5x+6y)}{8x+5y} &= 0\\
\end{align*}$$
And if one substitutes in
$$\begin{align*}
x=\frac{3}{\sqrt{2}}\\
y=\frac{-3}{\sqrt{2}}
\end{al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1272919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove that the integral of $\sin^2(x)/(5+3\cos(x))$ from $0$ to $2\pi$ is $2\pi/9$ I'm not really unsure of how to approach this problem. I was thinking of reparametrizing the sin and the cos to its exponential form but I realize that it becomes even messier and leads sort of nowhere. There are no singularities for th... | Suppose we seek to evaluate
$$\int_0^{2\pi} \frac{\sin^2 x}{5+3\cos x} dx.$$
Put $z = \exp(ix)$ so that $dz = i\exp(ix) \; dx$ and hence
$\frac{dz}{iz} = dx$ to obtain
$$\int_{|z|=1}
\frac{(z-1/z)^2/4/(-1)}{5+3/2(z+1/z)} \frac{dz}{iz}
\\ = -\int_{|z|=1}
\frac{(z-1/z)^2}{20+6(z+1/z)} \frac{dz}{iz}
\\ = -\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1274726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
How does $-\frac{1}{x-2} + \frac{1}{x-3}$ become $\frac{1}{2-x} - \frac{1}{3-x}$ I'm following a solution that is using a partial fraction decomposition, and I get stuck at the point where $-\frac{1}{x-2} + \frac{1}{x-3}$ becomes $\frac{1}{2-x} - \frac{1}{3-x}$
The equations are obviously equal, but some algebraic mani... | Each of the terms was multiplied by $\frac{-1}{-1}$, which is really equal to $1$, so it's a "legal" thing to do:
$-\dfrac{1}{x - 2} + \dfrac{1}{x - 3}$
$ = -\dfrac{(-1)1}{(-1)(x - 2)} + \dfrac{(-1)1}{(-1)(x - 3)}$
$ = -\dfrac{-1}{2 - x} + \dfrac{-1}{3 - x}$
$ = \dfrac{1}{2 - x} - \dfrac{1}{3 - x} $
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1275071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 7,
"answer_id": 0
} |
How to compute $\sum_{n=0}^{\infty}{\frac{3^n(n + \frac{1}{2})}{n!}}$? I have to compute the series $\displaystyle\sum_{n=0}^{\infty}{\frac{3^n(n + \frac{1}{2})}{n!}}$.
$$\displaystyle\sum_{n=0}^{\infty}{\frac{3^n(n + \frac{1}{2})}{n!}} = \sum_{n=0}^{\infty}{\frac{3^n\frac{1}{2}}{n!}} + \sum_{n=0}^{\infty}{\frac{3^nn}... | $$\begin{align}
\sum_{n=0}^\infty\frac{3^nn}{n!} & =0+\sum_{n=1}^\infty\frac{3^n}{(n-1)!}\\
& =\sum_{n=0}^\infty\frac{3^{n+1}}{n!}\\
& =3e^3.\\
\end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1275769",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
} |
Not understanding solution to $\large \int_{|z-2|=2} \frac {5z+7}{z^2+2z-3}dz$ computation Not understanding solution to $\large \int_{|z-2|=2} \frac {5z+7}{z^2+2z-3}dz$ computation.
What was shown in class: $\large \int_{|z-2|=2} \frac {5z+7}{z^2+2z-3}dz=\large \int_{|z-2|=2} \frac {5z+7}{(z+3)(z-1)}=\cdots= \int_{|... | Let $D=\{z: |z-2|\leq2\}$, thus $\partial D=\{|z-2|=2\}$. Since $f(z)=2/(z+3)$ is clearly analytic in $D$, then by Cauchy's Theorem $$ \int_{\partial D} f(z)dz=0.$$
However, if $g(z)=3/(z-1)$, since $1 \in D$, by Cauchy's Integral Formula, letting $g(z)=h(z)/(z-1)$ with $h\equiv 3$ we get
$$
\int_{\partial D} g(z)dz=\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1276304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Limit of functions involving trigonometry as n approaches infinity
By graphing these functions, I know that P(n) approaches pi as n tends towards infinity. However, is there a mathematical way for proving this?
I am doing a maths exploration on Archimedes' approximation of pi and those are the formulas that I derived ... | I think that Taylor series could be a simple solution.
Considering that, for small $x$, $$\tan(x)=x+\frac{x^3}{3}+O\left(x^4\right)$$ $$n \tan\big(\frac{\pi}n\big)=n \Big(\frac{\pi }{n}+\frac{\pi ^3}{3 n^3}+\cdots)=\pi+\frac{\pi ^3}{3 n^2}+\cdots$$ For the second (without using the good hint user222031 provided in com... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1277944",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Minimum value of trigonometric equation Find the minimum value of the expression $$y=\frac{16-8\sin^{2} 2x +8\cos^{4} x}{\sin^{2} 2x} .$$ When I convert the expression completely into $2x$, cross multiply and make the discriminant of the quadratic equation greater than $0$, I get the minimum value $-\infty$. I know it ... | $$y=\frac{16-8\sin^{2} 2x +8\cos^{4} x}{\sin^{2} 2x}$$
or $$y=\frac{16}{4}*\sec^2 x \csc^2 x-8 +\frac{8}{4}\frac{\cos^{4} x}{\sin^2 x* \cos^2 x}$$
or
$$y=4(\tan^2 x +1)(\cot^2 x +1)-8+2 \cot^2 x$$
$$y=4(\tan^2 x + \cot^2 x + 2)-8+2 \cot^2 x$$
Which after simplifying gives,
$$y=4\tan^2 x + 6\cot^2 x$$
Edit: as us... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1280639",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Computing a limit similar to the exponential function I want to show the following limit:
$$
\lim_{n \to \infty}
n
\left[
\left( 1 - \frac{1}{n} \right)^{2n}
- \left( 1 - \frac{2}{n} \right)^{n}
\right]
= \frac{1}{e^{2}}.
$$
I got the answer using WolframAlpha, and it seems to be correct numerically, but I ... | I would do the transform
$$\left(1-\frac{1}{n}\right)^{2n}=e^{2n\log\left(1-\frac{1}{n}\right)}$$
then use the second order Taylor expansion
$$\log(1+x)\approx x-\frac{x^2}{2}$$
and similarly for the other term, obtaining
$$
n\left(e^{-2-\frac{1}{n}}-e^{-2-\frac{2}{n}}\right)=e^{-2-\frac{2}{n}}\cdot\frac{e^{\frac{1}{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1282610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
find equality between linear spans $$U = Sp\{(2,5,-4,-10), (1,1,1,1),(1,0,3,5), (0,2,-4,-8)\}$$
$$ W = Sp\{(1,-2,7,13), (3,1,7,11), (2,1,4,6) \}$$
two questions:
*
*prove that $U = W$
*find the values of the $a \in \mathbb{R}$ where the vector $v=(a,a-6,4a-3,6a-1)$ belongs to $U$
i thought about solving it with a... | Let
\begin{align*}
A &=
\begin{bmatrix}
2&1&1&0\\
5&1&0&2\\
-4&1&3&-4\\
-10&1&5&-8
\end{bmatrix}
&
B&=
\begin{bmatrix}
1&3&2\\
-2&1&1\\
7&7&4\\
13&11&6
\end{bmatrix}
\end{align*}
Note the equation $AX=B$ has a (non-unique!) solution
$$
X=
\begin{bmatrix}
-1&-2/3&-1/3\\
3&13/3&8/3\\
0&0&0\\
0&0&0
\end{bmatrix}
$$
Also... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1284045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Simplifying quartic complex function in terms of $\cos nx$ $$z= \cos(x)+i\sin(x)\\
3z^4 -z^3+2z^2-z+3$$
How would you simplify this in terms of $\cos(nx)$?
| $$z= \cos(x)+i\sin(x)=e^{xi}$$
$$3z^4 -z^3+2z^2-z+3$$
So we got:
$$3(e^{xi})^4-(e^{xi})^3+2(e^{xi})^2-(e^{xi})+3=$$
$$3e^{4xi}-e^{3xi}+2e^{2xi}-e^{xi}+3=$$
$$(e^{ix}+e^{2ix}+1)(-4e^{ix}+3e^{2ix}+3)=$$
$$2e^{2ix}(2\cos(x)+1)(3\cos(x)-2)=$$
$$2e^{2ix}\left(6\cos^2(x)-\cos(x)-2\right)=$$
$$(2(\cos(2x)+\sin(2x)i))\left(6\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1286191",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Probability distribution of number of columns that has two even numbers in a chart
We distribute numbers $\{1,2,...,10\}$ in random to the following chart:
Let $X$ be the number of columns that has two even numbers.
What is the distribution of $X$?
My attempt:
$|\Omega|=10!$
$P(x=0)=\frac {(5!)^2\cdot 2 ^5}{10!}=\fr... | Let us grind it out.
There are five odd and five even. So we might as well assume that we use five $0$'s and five $1$'s. Let $X$ be the number of columns with two $0$'s. We want the probability distribution of $X$.
The random variable $X$ can only take on the values $0$, $1$, and $2$. So we really have only two proba... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1286415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to prove that $2^{n+2}+3^{2n+1}$ is divisible by 7 using induction? I want to prove that $2^{n+2}+3^{2n+1}$ is divisible by 7 using induction. My first step is replace $n$ with $1$.
*
*$2^{1+2}+3^{2(1)+1}$
*$2^3+3^3$
*$8+27$
*$35 = 7\times 5$
The next step is assume that $2^{n+2}+3^{2n+1}$ is divisible by 7... | Let $$f(n)=2^{n+2}+3^{2n+1}$$ so that $$f(n+1)=2^{n+3}+3^{2n+3}$$
Then
$$ \begin{align} f(n+1)-f(n) &= 2^{n+3}+3^{2n+3}-2^{n+2}-3^{2n+1} \\
&= 2^{n+2} \left(2-1 \right)+3^{2n+1}\left(3^2 - 1 \right) \\
&= 2^{n+2} + 8\left(3^{2n+1}\right) \\
&= 2^{n+2} + 3^{2n+1} + 7\left(3^{2n+1}\right) \end{align} $$
We assumed that $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1286522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 3
} |
Inequality for sides and height of right angle triangle
Someone recently posed the question to me for the above, is c+h or a+b greater, without originally the x and y lengths. I used this method: (mainly pythagorus)
$a^2+b^2=c^2=(x+y)^2=x^2+y^2+2xy$
$a^2=x^2+h^2$ and $b^2=y^2+h^2$
therefore $x^2+h^2+y^2+h^2=x^2+y^2+2x... | It is not true for all triangles.
From the law of cosines, $c^2 = b^2+ a^2 - 2ab \cos \gamma$, we can see that it is true when the triangle is obtuse.
On the other hand, it is not true for a triangle where all sides are equal.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1286755",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
How to find differential equation $$\frac{dy}{dx}-8x=2xy^2\quad y=0\,x=1$$
I separated $x$ and $y$.
\begin{align*}
\color{red}{\frac{dy}{y^2}}&=\color{red}{2x+8x dx}\\
\frac{dy}{y^2}&=\color{red}{10x dx}\\
\color{red}{\ln y^2} &= 5x^2\\
y^2&=Ae^{5x^2}
\end{align*}
When I plug $y$ and $x$ in, i get $A=0$.
I think I did... | $y′=2xy^2+8x⇒y′=x(2y^2+8).$ By separation of variables you find that $$y(x)=2\tan(2(2c_1+x^2))$$ and with the boundary condition $y(1 )=0$ the solution is $$y(x)=2\tan(2(x^2-1))$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1287070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Cannot understand an Integral $$\displaystyle \int _{ \pi /6 }^{ \pi /3 }{ \frac { dx }{ \sec x+\csc x } } $$
I had to solve the integral and get it in this form.
My attempt:
$$\int _{ \pi /6 }^{ \pi /3 }{ \frac { dx }{ \sec x+\csc x } } $$
$$=\int _{\frac{\pi}{6}}^{ \frac{\pi}{3}} \dfrac{\sin x \cos x }{ \sin x+\cos ... | Here is an alternate method you could use:
Multiply $\displaystyle\int\frac{\sin x\cos x}{\sin x+\cos x}dx$ on the top and bottom by $\cos x-\sin x$ to get
$\hspace{.6 in}\displaystyle\int\frac{\cos^2x\sin x}{2\cos^2 x-1}dx-\int\frac{\sin^2x\cos x}{1-2\sin^2 x}dx$.
Now substitute $u=\cos x$ in the first integral and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1288034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
About right identity which is not left identity in a ring Let $S$ be the subset of $M_2(\mathbb{R})$ consisting of all matrices of the form
$\begin{pmatrix}
a & a \\
b & b
\end{pmatrix}$
The matrix $\begin{pmatrix}
x & x \\
y & y
\end{pmatrix}$ is right identity in $S$ if and only if $x+y=1$. Fine, I can see that.
B... | To prove that $S$ contains no left identity, let an arbitrary
element
$
A
=
\begin{bmatrix}
x & x \\
y & y
\end{bmatrix}
\in S $ be given. Now, either $x = 0$ or $x \ne 0$. If $x = 0$,
then note that
$
\begin{bmatrix}
1 & 1 \\
0 & 0
\end{bmatrix}
\in S
$,
but
\begin{equation*}
A
\begin{bmatrix}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1293167",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
How to find the integral with $\sqrt [ 3 ]{ x } +\sqrt [ 4 ]{ x } $ in the denominator? How to evaluate $$\int { \frac { 1 }{ \sqrt [ 3 ]{ x } +\sqrt [ 4 ]{ x } } } +\frac { \log { (1+\sqrt [ 6 ]{ x } ) } }{ \sqrt [ 3 ]{ x } +\sqrt { x } } dx$$ I'm not being able to make the right substitution.Help please!
| when you have so many fractional powers, to simplify things try to go for a substitution that can clear up all the powers (usually the LCM of all denominators of various powers will do the job). For example, here let $x=t^{12}$, then you get
\begin{align*}
\int {\frac{1}{\sqrt[3]{x} +\sqrt[4]{x}}} +\frac{\log{(1+\sqrt[... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1293267",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
how to parameterize the ellipse $x^2 + xy + 3y^2 = 1$ with $\sin \theta$ and $\cos \theta$ I am trying draw the ellipse $x^2 + xy + 3y^2 = 1$ so I can draw it. Starting from the matrix:
$$ \left[ \begin{array}{cc} 1 & \frac{1}{2} \\ \frac{1}{2} & 3 \end{array}\right]$$
I computed the eigenvalues $2 \pm \frac{1}{2}\sq... | You have $x^2 +xy + 3y^2=(x+1/2 y)^2+11/4 y^2=1$.
You can then take:
$$\begin{array}{lll}
\sin \theta & = & x+1/2 y\\
\cos \theta &= &\sqrt{11}/2 y
\end{array}$$
Which is equivalent to:
$$\begin{array}{lll}
y &= 2/\sqrt{11} \cos \theta\\
x &=\sin \theta - 1/\sqrt{11} \cos \theta
\end{array}$$
to get a parametrization l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1294442",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Find $\Big\{ (a,b)\ \Big|\ \big|a\big|+\big|b\big|\ge 2/\sqrt{3}\ \text{ and }\forall x \in\mathbb{R}\ \big|a\sin x + b\sin 2x\big|\le 1\Big\}$
Find all (real) numbers $a $ and $b$ such that $|a| + |b| \ge 2/\sqrt{3} $ and for any $x$ the inequality $|a\sin x + b \sin 2x | \le 1$ holds.
In other words, find the set... | using condition and $sin(2x)=2sinx\cdot cosx$ we have for $c\in (0,1]$
$|a+2b\sqrt{1-c^2}|\le \dfrac 1 c \\$
$|a-2b\sqrt{1-c^2}|\le \dfrac 1 c \Rightarrow \\$
$|a|+2|b|\sqrt{1-c^2}\le \dfrac 1 c \\$
$c=\dfrac {\sqrt3} 2 \Rightarrow |a|+|b|\le \dfrac 2 {\sqrt3} \Rightarrow |a|+|b|=\dfrac 2 {\sqrt3} \\ $
so we have
$\df... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1294971",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Trigonometric equation $\sec(3\theta/2) = -2$ - brain dead Find $\theta$ with $\sec(3\theta/2)=-2$ on the interval $[0, 2\pi]$. I started off with $\cos(3 \theta/2)=-1/2$, thus $3\theta/2 = 2\pi/3$, but I don't know what to do afterwards, the answer should be a huge list of $\theta$s, which I cannot seem to get.
| Your procedure is correct, except you forgot to include the many other values of $\cos^{-1}(-\frac12)$.
$\cos{x} = -\frac12$ for $x = \frac{2\pi}{3}\pm2\pi n, \frac{4\pi}{3}\pm2\pi n$, where $n$ is an integer.
So you would have:
$$ \frac{3\theta}{2} = \frac{2\pi}{3} \to \theta = \frac{4\pi}{9} $$
$$ \frac{3\theta}{2} =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1295926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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The lines $x+2y+3=0$ , $x+2y-7=0$ and $2x-y+4=0$ are sides of a square. Equation of the remaining side is? I found out the area between parallel lines as $ \frac{10}{\sqrt{5}} $ and then I used
$ \frac{|\lambda - 4|}{\sqrt{5}} = \frac{10}{\sqrt{5}} $ to get the values as $-6$ and $14$ . I am getting the final equatio... | The first two lines are parallel to each other, and so you are looking for a line parallel to the third one. You have
*
*$x+2y+3=0$
*$x+2y-7=0$
*$2x+(-1)y+4=0$
*$2x+(-1)y+K=0$
where $K$ is an unknown coefficient to complete the square.
The distance between the first and second line (square side) is
$$ d_{12} =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1296690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Using Gauss elimination to check for linear dependence I have been trying to establish if certain vectors are linearly dependent and have become confused (in many ways). when inputting the vectors into my augmented matrix should they be done as columns or as rows ?
eg if my vectors are [ 4 -1 2 ], [-4 10 2]
I am looki... | If $ \: \: a \begin{pmatrix}
4 \\
2 \\
2
\end{pmatrix} $ + $ b \begin{pmatrix}
2 \\
3 \\
9
\end{pmatrix} $ = $ \begin{pmatrix}
0 \\
0 \\
0
\end{pmatrix} \: \:$ then $ \: \: \begin{pmatrix}
4a+2b \\
2a+3b\\
2a+9b
\end{pmatrix} $ = $ \begin{pmatrix}
0 \\
0 \\
0
\end{pmatrix}$.
So we hav... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1297437",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
The four straight lines given by the equation $12x^2+7xy-12y^2 =0$ and $12x^2+7xy-12y^2-x+7y-1=0$ lie along the side of the? I know these equations are called general equation of second degree and also represent a pair of straight lines. I could extract lines from the equation $$12x^2+7xy-12y^2 =0 $$ (these are $$ 3x+4... | In order to avoid the cumbersome calculations, assume that the lines are: $y=m_{1}x+c_{1}$ & $y=m_{2}x+c_{2}$ Now, the quadratic equation of pair of the lines is given as
$$(m_{1}x-y+c_{1})(m_{2}x-y+c_{2})=0 $$$$\implies m_{1}m_{2}x^2-(m_{1}+m_{2})xy+y^2+(m_{1}c_{2}+m_{2}c_{1})x-(c_{1}+c_{2})y+c_{1}c_{2}=0 \tag 1$$ Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1298343",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
How do I find the factorial of a decimal How do I find the following:
$$(0.5)!(-0.5)!$$
Can someone help me step by step here?
| Factorial of any real number $n$ is defined by Gamma function as follows:
$$\Gamma (n) = (n-1)!$$
$$\quad \Rightarrow ( \dfrac{1}{2} )! ( -\dfrac{1}{2} ) ! = ( \dfrac{3}{2}-1 ) ! ( \dfrac{1}{2}-1 ) ! = \Gamma ( \dfrac {3} {2} ) \Gamma ( \dfrac {1}{2} )$$
It is also known that:
$$\Gamma {(1+z)} = z\Gamma {(z)}$$
$$\quad... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1304344",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Finding basis made of uninvertible matrices Let there be transformation $T: \mathbb R_3[X] \rightarrow M_{2 \times 2}(\mathbb R)$, $T(ax^3+bx^2+cx+d)=\left[ \begin{matrix}
a+d & b-2c \\
a+b-2c+d & 2c-b \\
\end{matrix} \right] $
Find a basis of $Im(T)$ made of non-invertible matrices.
So... | $sp\{\left[ \begin{matrix}
1 & 0 \\
1 & 0 \\
\end{matrix} \right], \left[ \begin{matrix}
0 & 1 \\
1 & -1 \\
\end{matrix} \right] \} =
sp\{\left[ \begin{matrix}
\frac{1}{2} & 0 \\
\frac{1}{2} & 0 \\
\end{matrix} \right], \le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1305652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to evaluate $\lim\limits_{n\to\infty}\frac{1}{n}((x+\frac{a}{n})^2+(x+\frac{2a}{n})^2+...+(x+\frac{(n-1)a}{n})^2)$ I don't know how to transform the expression $\frac{1}{n}((x+\frac{a}{n})^2+(x+\frac{2a}{n})^2+...+(x+\frac{(n-1)a}{n})^2)$
The solution, after transformation is
$\frac{n-1}{n}x^2+2\frac{1+...+(n-1)}{... | We have $$\lim\limits_{n\to\infty}\frac{1}{n}((x+\frac{a}{n})^2+(x+\frac{2a}{n})^2+...+(x+\frac{(n-1)a}{n})^2)=\lim\limits_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n-1}(x+\frac{k}{n}a)^2$$ and so we have that $$\lim\limits_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n-1}(x+\frac{k}{n}a)^2=\int_{0}^{1}(x+ta)^2 dt=x^2+\frac{a^2}{3}+ax+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1306361",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Integration by Parts Problem: Help in understanding why a part of it equals 0 $$4I= \int_0^{\infty} \frac{4x^3 +\sin(3x)-3\sin x}{x^5} \ \mathrm{d}x $$ $$=\frac{-1}{4} \underbrace{\left[\frac{4x^3+\sin(3x)- 3 \sin x}{x^4} \right]_0^{\infty}}_{=0} +\frac{1}{4} \int_0^{\infty} \frac{12x^2 -3\cos x +3\cos(3x)}{x^4} \ \mat... | Let's evaluate
$${\left[\frac{4x^3+\sin(3x)- 3 \sin x}{x^4} \right]_0^{\infty}} =$$
$$\lim_{x\to \infty}(\frac{4x^3+\sin(3x)- 3 \sin x}{x^4} )-\lim_{x\to 0}(\frac{4x^3+\sin(3x)- 3 \sin x}{x^4} )$$
take a look on $$\lim_{x\to \infty}(\frac{4x^3+\sin(3x)- 3 \sin x}{x^4} )$$
$sin(3x)$ and $3sin(x)$ are a bounded functio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1307972",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the most unusual proof you know that $\sqrt{2}$ is irrational? What is the most unusual proof you know that $\sqrt{2}$ is irrational?
Here is my favorite:
Theorem: $\sqrt{2}$ is irrational.
Proof:
$3^2-2\cdot 2^2 = 1$.
(That's it)
That is a corollary of
this result:
Theorem:
If $n$ is a positive integer... | Consider the linear application $A:\mathbb{R}^2\to \mathbb{R}^2$ given by $$A=\begin{pmatrix}
-1&2 \\ 1&-1
\end{pmatrix} .$$ $A$ maps $\mathbb{Z}^2$ into itself and $V=\{y=\sqrt 2 x\}$ is an eigenspace relative to the eigenvalue $\sqrt 2-1$. But $A\mid_V$ is a contraction mapping, so $\mathbb{Z}^2\cap V=\emptyset$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1311228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "114",
"answer_count": 19,
"answer_id": 8
} |
prove that the expression $\frac{(3n)!}{(3!)^n}$ is integral for $n \geq 0$ My concept of real no. Is not very clear. Please also tell the logic behind the question.
The expression is true for 19, is it true for all the multiples?
| You want to prove $\frac{(3n)!}{6^n}$ is an integer. Just use $\frac{3n!}{6^n}=\frac{1\cdot2\cdot3}{6}\frac{4\cdot 5\cdot 6}{6}\frac{7\cdot8\cdot 9}{6}\dots \frac{(3n-2)(3n-1)(3n)}{6}$ and each fraction is an integer since $k(k+1)(k+2)$ is always a multiple of $2$ and of $3$ since three consecutive integers always cont... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find probability of exactly one $6$ in first ten rolls of die, given two $6$s in twenty rolls I am trying to calculate the probability that, when rolling a fair die twenty times, I roll exactly one $6$ in the first ten rolls, given that I roll two $6$s in the twenty rolls.
My thoughts
Let $A = \{\text {Exactly one 6 in... | I took a different approach to the question. Suppose B. There are three ways to get two 6's in twenty rolls:
*
*$B_1$: Both 6's come in the first 10 rolls. There are $\begin{pmatrix} 10 \\ 2\end{pmatrix} = 45$ ways for this to happen.
*$B_2$: One 6 comes in the first 10 rolls, and the second comes in the next 10 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1312058",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Modular arithmetic , calculate $54^{2013}\pmod{280}$.
How do you calculate: $54^{2013}\pmod{280}$?
I'm stuck because $\gcd(54,280)$ is not $1$. Thanks.
| As $280=2^3\cdot5\cdot7$
let us start with $54^{2013-3}\pmod{\dfrac{280}{2^3}}$ i.e., $54^{2010}\pmod{35}$
Now $54\equiv-1\pmod5\implies54^2\equiv(-1)^2\equiv1$
and $54\equiv-2\pmod7\implies54^3\equiv(-2)^3\equiv-1,54^6\equiv(-1)^2\equiv1$
$\implies54^6\equiv1\pmod{35}$
As $2010\equiv0\pmod6,54^{2010}\equiv54^0\pmod{35... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1312311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Integer partitioning Suppose we have an integer $n$. I we want to partition the integer in the form of $2$ and $3$ only; i.e., $10$ can be partitioned in the form $2+2+2+2+2$ and $2+2+3+3$.
So, given an integer, how to calculate the total number of ways of doing such partitions and how many $2$'s and $3$'s are there i... | For even $n$, the number of $3$ in each partition is even. So, let $2k$ be the largest number of $3$ in the partition of even $n$, i.e.
$$3\cdot 2k\le n\lt 3(2k+2)\Rightarrow k\le \frac{n}{6}\lt k+1\Rightarrow k=\left\lfloor\frac n6\right\rfloor.$$
Hence, the number of partitions of $\color{red}{\text{even}\ n}$ is $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1313630",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What two numbers when multiplied gives $-25$ and when added, $-10$? Factors of $25$ are; $1, 5, 25$
$$5 \times 5 = 25$$
$$5 + 5 = 10$$
$$-5 \times 5 = -25$$
$$-5 + 5 = 0$$
How can I solve this?
Thanks
| Solve: $$\begin{align}
xy &=-25 \\
x+y &= -10
\end{align}
$$
In particular, substituting $y=-10-x = -(10+x)$ in the first equation, you get $$x^2+10x -25 = 0 $$
which is an equation you should know how to handle. The two solutions are $x=-5+\sqrt{50} = 5(\sqrt{2}-1)$ and $x=-5(\sqrt{2}+1)$, from which you get respectiv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1316134",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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solving difficult complex number proving if $z= x+iy$ where $y \neq 0$ and $1+z^2 \neq 0$, show that the number $w= z/(1+z^2)$ is real only if $|z|=1$
solution :
$$1+z^2 = 1+ x^2 - y^2 +2xyi$$
$$(1+ x^2 - y^2 +2xyi)(1+ x^2 - y^2 -2xyi)=(1+ x^2 - y^2)^2 - (2xyi)^2$$
real component
$$(1+ x^2 - y^2)x - yi(2xyi) = x + x^3... | $$\dfrac Z{1+Z^2}=\dfrac{x+iy}{1+x^2-y^2+2xyi}$$
$$=\dfrac{(x+iy)(1+x^2-y^2-2xyi)}{(1+x^2-y^2)^2+(2xy)^2}$$
We need $y(1+x^2-y^2)-x(2xy)=0\implies y(1+x^2-y^2-2x^2)=0\iff x^2+y^2=1$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Limit of $\dfrac{(1+4^x)}{(1+3^x)}$? I don't remember how to find the limit in this case. I take $x$ towards $+\infty$.
$\lim\limits_{x\to \infty} \dfrac{1+4^x}{1+3^x}$
I do not know where to start. I would instinctively say that $1$ can't be right because $4^x$ goes faster than $3^x$ and thus one would move towards ... | $$\lim_{x\rightarrow \infty}\dfrac{(1+4^x)}{(1+3^x)}=$$
$$\lim_{x\rightarrow \infty}\frac{1}{1+3^x}+\lim_{x\rightarrow \infty}\frac{4^x}{1+3^x}=$$
$$\frac{1}{\lim_{x\rightarrow \infty} 1+3^x}+\lim_{x\rightarrow \infty}\frac{4^x}{1+3^x}=$$
$$\frac{1}{\infty}+\lim_{x\rightarrow \infty}\frac{4^x}{1+3^x}=$$
$$0+\lim_{x\rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1319950",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show if $\sum_{n=1}^{\infty} \frac{(-1)^n}{1+\sqrt{n}}$ is absolute convergent or divergent Show if $\sum_{n=1}^{\infty} \frac{(-1)^n}{1+\sqrt{n}}$ is absolute convergent or divergent
First i subbed numbers in
$$\lim_{n \to \infty} \frac{(-1)^n}{1+\sqrt{n}} = \frac{-1}{1+\sqrt{1}} + \frac{1}{1+\sqrt{2}} - \frac{-1}{1+\... | Let $a_n=\frac{(-1)^n}{1+\sqrt{n}}$.
$$ a_{2n}+a_{2n+1} = \frac{1}{1+\sqrt{2n}}-\frac{1}{1+\sqrt{2n+1}} = \frac{\sqrt{2n+1}-\sqrt{2n}}{(1+\sqrt{2n})(1+\sqrt{2n+1})} = \sqrt{2n} \frac{\sqrt{1+\frac{1}{2n}}-1}{(1+\sqrt{2n})(1+\sqrt{2n+1})} \sim \frac{n^{-\frac{3}{2}}}{4\sqrt{2} } = O\left(\frac{1}{n^{\frac{3}{2}}}\rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1322459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
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How to solve this inequality, with the hypothesis more complicated than the conclusion? Given $x,y,z \in \mathbb{R}$ and $x,y,z>2,$ I want to show that if,
$$\frac{1}{x^2-4}+\frac{1}{y^2-4}+\frac{1}{z^2-4} = \frac{1}{7}$$
then,
$$\frac{1}{x+2} + \frac{1}{y+2} + \frac{1}{z+2} \leq \frac{3}{7}.$$
I follow the solution h... | Here is an adaptation of pi37 answer in your given link.
Note
$$\sum_{cyc}\dfrac{x^2+25}{x^2-4}=\sum_{cyc}\dfrac{x^2-4+29}{x^2-4}=3+\dfrac{29}{7}=\dfrac{50}{7}$$
and use AM-GM $x^2+25\ge 10x$.so
$$\sum_{cyc}\dfrac{x^2+25}{x^2-4}\ge\dfrac{10x}{x^2-4}\Longrightarrow\sum_{cyc}\dfrac{x}{x^2-4}\le \dfrac{5}{7}$$
and note
$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 2
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How to find out the greater number from $15^{1/20}$ and $20^{1/15}$? I have two numbers $15^{\frac{1}{20}}$ & $20^{\frac{1}{15}}$.
How to find out the greater number out of above two?
I am in 12th grade. Thanks for help!
| Well raise both numbers to the power of $20$
That is
$$\large{(15^\frac{1}{20})^{20} = 15^\frac{20}{20} = 15}$$
Now $$\large{(20^\frac{1}{15})^{20} = 20^\frac{20}{15} = 20^\frac{4}{3} = 20^{1.333..}}$$
which is greater ? $\large{15}$ or $\large{20^{1.333...}}$
Clearly , it is $\large{20^{1.333..}}$ because $\large{20^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1327118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Given $x^2 + 4x + 6$ as factor of $x^4 + ax^2 + b$, then $a + b$ is I got this task two days ago, quite difficult for me, since I have not done applications of Vieta's formulas and Bezout's Theorem for a while. If can someone solve this and add exactly how I am supposed to use these two theorem's on this task, I would ... | \begin{align*}
x^4+ax^2+b & = x^2\color{blue}{(x^2+4x+6)}-4x\color{blue}{(x^2+4x+6)}+\\
&(a+10)\color{blue}{(x^2+4x+6)}-\color{red}{(4a+16)(x)+(b-6a-60)}
\end{align*}
For $x^2+4x+6$ to divide the given polynomial.We need $4a+16=0$ and $b-6a-60=0$. Thus $a=-4$ and $b=36$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1328210",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 2
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Solving a complex number inequality involving absolute values. Here is the relevant paragraph (from "Complex numbers from A to Z" by Titu Andreescu and Dorin Andrica) :
Original question : How does $\left | 1+z \right |=t$ imply $\left | 1-z+z^2 \right |=\sqrt{\left | 7-2t^2 \right |}$?
(I checked for $z=i$ , it seems... | Since $|z|$, we can write $$ z= \cos\theta+ \sin\theta$$
Now we can derive useful properties for such $z$:
$1+z = 1 + \cos\theta + i\sin\theta = 1 + 2\cos^2\frac{\theta}{2} - 1 + 2i\sin\frac{\theta}{2}\cos\frac{\theta}{2} = 2\cos\frac{\theta}{2}(\cos\frac{\theta}{2} + i \sin\frac{\theta}{2})$
Thus, $|1+z| = 2|\cos\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1329259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Divisibility of a polynomial by another polynomial I have this question:
Find all numbers $n\geq 1$ for which the polynomial $x^{n+1}+x^n+1$ is divisible by $x^2-x+1$. How do I even begin?
So far I have that $x^{n+1}+x^n+1 = x^{n-1}(x^2-x+1)+2x^n-x^{n-1}+1,$ and so the problem is equivalent to finding $n$ such that ... | Hint: The roots of $x^2-x+1=0$ are $e^{i\pi/3}$ and $e^{-i\pi/3}$. If we show that one (and therefore the other) cannot be a root of $x^{n+1}+x^n+1$, then we will know that $x^2-x+1$ cannot divide $x^{n+1}+x^n+1$.
There are $3$ cases to examine: (i) $n$ is of the form $3k+2$; (ii) $n$ is of the form $3k+1$; and (iii) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1329695",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Find the cubic equation of $x=\sqrt[3]{2-\sqrt{3}}+\sqrt[3]{2+\sqrt{3}}$ Find the cubic equation which has a root $$x=\sqrt[3]{2-\sqrt{3}}+\sqrt[3]{2+\sqrt{3}}$$
My attempt is
$$x^3=2-\sqrt{3}+3\left(\sqrt[3]{(2-\sqrt{3})^2}\right)\left(\sqrt[3]{(2+\sqrt{3})}\right)+3\left(\sqrt[3]{(2-\sqrt{3})}\right)\left(\sqrt[3]{(... | $$x^3=(r+s)^3=r^3+3r^2s+3rs^2+s^3=r^3+s^3+3rs(r+s)=4+3x,$$
because $r^3+s^3=4$ and $rs=1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1331417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 8,
"answer_id": 0
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Maximum value of trigonometric expression If
$r=3+\tan c \tan a, \quad
q=5+\tan b \tan c, \quad
p=7+\tan a \tan b$
Provided $a,b,c$ are positive and
$a+b+c=\dfrac{\pi}2$
Find the maximum value of
$\sqrt p + \sqrt q + \sqrt r$ .
| Let $x = \tan a \tan b, \; y = \tan b \tan c, \; z = \tan c \tan a$, then $x, y, z > 0$ and $x+y+z=1$.
We need to now maximize $\sqrt{3+x}+\sqrt{5+y}+\sqrt{7+z}$. As $\sqrt t$ is concave, using Karamata's inequality and $(3+x, 5+y, 7+z) \succ (3+x+y+z, 5, 7) = (4, 5, 7)$, we have $\sqrt{3+x}+\sqrt{5+y}+\sqrt{7+z} \le ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1332293",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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About the "Cantor volume" of the $n$-dimensional unit ball A simple derivation for the Lebesgue measure of the euclidean unit ball in $\mathbb{R}^n$ follows from computing
$$ \int_{\mathbb{R}^n}e^{-\|x\|^2}\,dx $$
in two different ways. See, for instance, Keith Ball, An Elementary Introduction to Modern Convex Geometry... | HINT:
For $\mu(x)$ the Cantor measure supported on the Cantor set $\subset [0,1]$ we have the change of variable formula:
$$\int f(x)\, d\mu(x) = \frac{1}{2} \int f(1/3 x)\, d\mu(x) + \frac{1}{2} \int f(1/3 x + 2/3)\, d \mu(x)$$
analogous to $\int_0^1 f(x)\, dx =\frac{1}{2} \int_0^1 f(1/2 x)\, d x + \frac{1}{2} \int_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1332996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 1,
"answer_id": 0
} |
Derivative and graph mismatch Using the implicit function $(x^2+y^2-1)^3=x^2y^3$ it can be shown that $y'=\frac{2xy^3-6x(x^2+y^2-1)^2}{6y(x^2+y^2-1)^2-3x^2y^2}$ but when I evaluate it for the point (1,0) I get $y'(1,0)=\frac{0}{0}$ even though the slope of the tangent line is 2 at that point.
Any ideas?
Garth
| Instead $(x^2+y^2-1)^3=x^2y^3$ we can take $x^2+y^2-1=x^{2/3}y$ near of $(1,0)$, hence
\begin{align*}
\left(y-\frac{1}{2}x^{2/3}\right)^2&=1-x^2+\frac{x^{4/3}}{4}\\
y&=-\left(1-x^2+\frac{x^{4/3}}{4}\right)^{1/2}+\frac{x^{2/3}}{2}\\
y'&=-\frac{1}{2}\left(1-x^2+\frac{x^{4/3}}{4}\right)^{-1/2}\left(-2x+\frac{x^{1/3}}{3}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1334031",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to evaluate $\lim_{n\to\infty} a_n$ If $a_1=1$ and $a_{n+1}=\frac{4+3a_n}{3+2a_n}$,$n\geq1$, then how to prove that $a_{n+2}>a_{n+1}$ and if $a_n$ has a limit as ${n\to\infty}$ then how to evaluate $\lim_{n\to\infty} a_n$ ?
| Consider the function $f(x) = \frac{4+3x}{3+2x} = \frac32 - \frac{1}{2(3+2x)}$. It is easy to check.
*
*$f(x)$ is a strictly increasing function from $[0,\infty)$ to $[0,\infty)$.
*$\sqrt{2}$ is a fixed point for $f(x)$.
So start from any number $a \in (0,\sqrt{2})$, if we construct a sequence $a_n$ by
$$a_{n} = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1334866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Prove or disprove that $8c+1$ is square number.
Let $a,b,c$ be positive integers, with $a-b$ prime, and $$3c^2=c(a+b)+ab.$$ Prove or disprove that $8c+1$ is square number.
| Yes, that's true! Here we go: Write $b=a-p$. Then, if you write the expression as a quadratic equation in $c$, you get $3c^2-(2a-p)c+ap-a^2=0.$ Now, by the quadratic formula, see that since we want $c$ to be integer, we must have $(2a-p)^2-12(ap-a^2)=x^2$ for some, say positive, integer $x$. Rewriting in a better way, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1335720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 2
} |
Find $x+y+z$ from the equation In the equation
$$\dfrac{37}{13}=2+\dfrac{1}{x+\dfrac{1}{y+\dfrac 1z}}$$
find the value of $x+y+z$
How can I know the value of three variables while there is only one equation?
| $$\frac{26+11}{13}=2+\frac{1}{\frac{13}{11}}=2+\frac{1}{1+\frac{1}{\frac{11}{2}}}=2+\frac{1}{1+\frac{1}{5+\frac{1}{2}}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1337653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Partial fraction of $\frac 1{x^6+1}$ Can someone please help me find the partial fraction of $$1\over{x^6+1}$$
?
I know the general method of how to find the partial fraction of functions but this seems a special case to me..
| \begin{align}
\frac 1{x^6+1}&=\frac 1{(x^2)^3+1}\\
&=\frac 1{((x^2)1+)((x^2)^2-(x^2)+1)} \\
&= \frac 1{(x^2+1)(x^4-x^2+1)} \\
&= \frac {Ax+B}{x^2+1}+\frac{Cx+D}{x^4-x^2+1}
\end{align}
So $$1=(x^4-x^2+1)(Ax+B)+(x^2+1)(Cx+D)$$
At $x=0$, the equation becomes $$1=B+D$$
At this point, equating the coefficients is the only ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 3
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solve $x^2 \equiv 24 \pmod {60}$ I need to solve $x^2 \equiv 24 \pmod {60}$
My first question which confuses me a lot -
isn't a (24 here) has to be coprime to n (60)???
most of the theorems requests that.
what i tried -
$ 60 = 2^2 * 3 * 5$
So I need to solve $x^2 \equiv 24$ modulo each one of $2^2, 3, 5$ so i get -... | You were correct.
$$x^2\equiv 24\pmod{\! 60}\iff \begin{cases}x^2\equiv 24\equiv 0\pmod{\! 3}\\ x^2\equiv 24\equiv 0\pmod{\! 4}\\ x^2\equiv 24\equiv 4\pmod{\! 5}\end{cases}$$
$$\iff \begin{cases}x\equiv 0\pmod{\! 3}\\ x\equiv 0\pmod{\! 2}\\ x\equiv \pm 2\pmod{\! 5}\end{cases}$$
If and only if at least one of the two ca... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Help with indefinite integration I am learning indefinite integration, yet am having problems understanding and recognizing where to substitute what. a good trick is to attempt convert algebraic expressions into trigonometric and vice versa. However, despite doing so, I am unable to solve the integral. For example, in... | Let
\begin{align}
I = \int \sqrt{\frac{1 - \sqrt{x}}{1+\sqrt{x}}} \, dx
\end{align}
and make the substitution $x = t^{2}$ to obtain
\begin{align}
I = 2 \, \int \sqrt{\frac{1-t}{1+t}} \, t \, dt.
\end{align}
Now let $t = \cos(2\theta)$ to obtain
\begin{align}
I &= -4 \, \int \sqrt{\frac{1- \cos(2\theta)}{1+\cos(2\theta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1338499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding the last digit of $7^n$, $n\ge 1$. I have noticed a cycle of 7,9,3,1. Meaning: $7^1\equiv 7\pmod {10}, 7^2\equiv 9\pmod {10}, 7^3\equiv 3\pmod {10},7^4\equiv 1\pmod {10}, 7^5\equiv 7\pmod {10}$ and so on. Therefore, if $n=4k+1$ the last digit is 7, If $n=4k+2$, the last digit is 9, If $n=4k+3$, the last digit i... | Take $n=4m+k$ where $0\leq k\leq 3$, then
$$7^{4m+k}=7^{4m}\cdot 7^k=(7^4)^m\cdot 7^k\equiv 1^m 7^k \equiv 7^k (mod 10).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1338595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Interesting variant of binomial distribution. Suppose i had $n$ Bernoulli trials with $X_{i}=1$ if the $i$th trial is a success and $X_{i}=-1$ if it is a failure each with probability $\frac{1}{2}$.
Then the difference between the number of successes and failures can be represented by the random variable $Y=|\sum_{i=1... | Let $S_{2n}$ be the number of successes in $2n$ trials. Then there are $2n - S_{2n}$ failures, with a difference of $|S_{2n} - (2n - S_{2n})| = |2S_{2n} - 2n|$. Now let $0 <k \leq 2n$ be an even number. Then
\begin{align*}
\mathbb{P}(|2S_{2n} - 2n| = k) &= \mathbb{P}(2S_{2n} - 2n = k) + \mathbb{P}(2S_{2n} - 2n = -k) \\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1338869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Find remainder when $777^{777}$ is divided by $16$
Find remainder when $777^{777}$ is divided by $16$.
$777=48\times 16+9$. Then $777\equiv 9 \pmod{16}$.
Also by Fermat's theorem, $777^{16-1}\equiv 1 \pmod{16}$ i.e $777^{15}\equiv 1 \pmod{16}$.
Also $777=51\times 15+4$. Therefore,
$777^{777}=777^{51\times 15+4}={(7... | Hint: $9^2~=~81~=~5\cdot16+1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1342082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Solving $\sin x=2\sin(2\pi/3-x)$ How can I solve the equation: $$\sin x=2\sin\left(\frac{2\pi}{3}-x\right)$$
Without using the formula:
$$\sin(a-b)=\sin a \cos b-\sin b \cos a$$?
Thanks.
| Consider the next picture
By Sine Law the equality $\frac{\sin x}{2} = \frac{\sin\left(2\pi/3-x\right)}{1}$ holds, then by using the Cosine Law we have
\begin{align*}
c^2&=5-4\cos\frac{\pi}{3}\\
&=3\\
c&=\sqrt{3}
\end{align*}
Now, from Sine Law, it follows.
\begin{align*}
\sin x&=\frac{2\sin \pi/3}{\sqrt{3}}\\
\sin x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1342309",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Sum of solutions of this exponential equations How to solve this :
$$x^{3-\log_{10}(x/3)}=900$$
I tried log on both sides and got nothing with exponent of $x$ and $3$.
| Go ahead and take the $\log_{10}$ on both sides:
$$3\log_{10}(x)-\log_{10}(x)^2+\log_{10}(x)\log_{10}(3)=\log_{10}(900).$$
Now solve the quadratic.
Let $y=\log_{10}(x).$ Then this quadratic is
$$y^2-(\log_{10}(3)+3)y+\log_{10}(900)=0.$$
Applying the quadratic formula, we get
$$y=\frac{3+\log_{10}(3)\pm\sqrt{\log_{10}(3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1343642",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove this inequality $\frac{1}{1+a}+\frac{2}{1+a+b}<\sqrt{\frac{1}{a}+\frac{1}{b}}$
Let $a,b>0$ show that
$$\dfrac{1}{1+a}+\dfrac{2}{1+a+b}<\sqrt{\dfrac{1}{a}+\dfrac{1}{b}}$$
It suffices to show that
$$\dfrac{(3a+b+3)^2}{((1+a)(1+a+b))^2}<\dfrac{a+b}{ab}$$
or
$$(a+b)[(1+a)(1+a+b)]^2>ab(3a+b+3)^2$$
this idea can't ... | if op go ahead , he will get from his last step :LHS-RHS$=(4a^4+4a^2+b^3-6a^2b)+(a^2b^2-3ab+b+a)+a^2b^3+ab^3+3a^3b^2+a^2b^2+ab^2+2b^2+3a^4b+a^3b+a^5+6a^3 >0 \iff$
$(4a^4+4a^2+b^3-6a^2b) \ge 0 \cap (a^2b^2-3ab+b+a)\ge 0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1344230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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$(x+y+z)^3-(y+z-x)^3-(z+x-y)^3-(x+y-z)^3=24xyz$? The question given is
Show that $(x+y+z)^3-(y+z-x)^3-(z+x-y)^3-(x+y-z)^3=24xyz$.
What I tried is suppose $a=(y+z-x),\ b=(z+x-y)$ and $c=(x+y-z)$ and then noted that $a+b+c=x+y+z$. So the question statement reduced to $(a+b+c)^3-(a^3+b^3+c^3)$. Then I tried to invoke t... | Observe that,
$\begin{align}(x+y+z)^3-(x+y-z)^3&=2z\left((x+y+z)^2+(x+y-z)^2)+(x+y)^2-z^2\right)\\&=2z\left(3(x+y)^2+z^2\right)\tag{1}\end{align}$
and,
$\begin{align}(x-y-z)^3-(x-y+z)^3&=-2z\left((x-y-z)^2+(x-y+z)^2)+(x-y)^2-z^2\right)\\&=2z\left(3(x-y)^2+z^2\right)\tag{2}\end{align}$
Therefore,
$(x+y+z)^3+(x-y-z)^3-(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1344314",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
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Hints on solving $y'=\frac{y}{3x-y^2}$
$$y'=\frac{y}{3x-y^2}$$
My attempt:
$$\frac{dy}{dx}=\frac{y}{3x-y^2}$$
$$dy\cdot(3x-y^2)=dx\cdot y$$
$$dy\cdot3x-dy\cdot y^2=dx\cdot y$$
Any direction?
I need hints please $\color{red}{not}$ a full answer
| Following the hint given by Chinny84, we write the equation as $$\dfrac{\mathrm{d}x}{\mathrm{d}y} = \frac{3x-y^2}{y} = \frac{3x}{y} -y$$
So $$\frac{\mathrm{d}x}{\mathrm{d}y} - \frac{3x}{y} = -y$$ which is a linear first order differential equation in $x$.
Spoiler: Answer below.
Using an integrating factor $$I = \exp\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1344864",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Unusual result to the addition
Question: Prove that
(666... to n digits)^2 + (888... to n digits)=(444... to 2n digits)
My way: I just proved the given equation for three values of n and written at the bottom.
"Since the equation satisfies for n=1, 2, and 3, the equation is true and hence proved."
Also I am seeing a ... |
Question: Prove that
$(666\dots \text{to $n$ digits})^2 + (888\dots \text{to $n$
> digits})=(444\dots \text{to $2n$ digits})$
By dividing by $4$, this is equivalent to
\begin{align}
(333\dots \text{to $n$ digits})^2
+ (222\dots \text{to $n$ digits})
&=(111\dots \text{to $2n$ digits})
\end{align}
Let $x=111\dots \tex... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1345786",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Find the sum$\pmod{1000}$
Find $$1\cdot 2 - 2\cdot 3 + 3\cdot 4 - \cdots + 2015 \cdot 2016 \pmod{1000}$$
I first tried factoring,
$$2(1 - 3 + 6 - 10 + \cdots + 2015 \cdot 1008)$$
I know that $\pmod{1000}$ is the last three digits.
$$= 2 - 6 + 12 - 20 + 30 - 42 + \cdots + 240$$
that is too complicated. Just hints ple... | The given series is: $$\sum_{i=1} ^{1008} (2i-1)(2i)-\sum_{i=1} ^{1007}(2i)(2i+1)$$ which is $$4\times\sum_{i=1} ^{1008} i^2-2\times \sum_{i=1} ^{1008} i-4\times \sum_{i=1} ^{1007} i^2+2\times \sum_{i=1} ^{1007} i$$ which, by cancelling out the common terms becomes, $$4\times 1008^2-2\times 1008=2\times 1008\times (2\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1347213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Can we obtain $f(y+x)=y+f(x)$ from $f(x^2+f(x)^2+x)=f(x)^2+x^2+f(x)$?
$\mathbb Z^+$ is the set of positive integers. Find all functions $f:\mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that
$$f(m^2+f(n))=f(m)^2+n\quad(\clubsuit)$$
Let $P(x,y)$ be the assertion: $f(x^2+f(y))=f(x)^2+y \; \forall x,y \in \mathbb{Z}^+.$
$... | Given
$$
f: \mathbb{Z} \rightarrow \mathbb{Z} \wedge f(m^2 + f(n)) = f(m)^2 + n.
$$
For a fixed $m$ and $n=x$ we get
$$
f(m^2 + f(x)) = f(m)^2 + x.
$$
In general we can write
$$
f(x) = \sum_k a_k x^k.
$$
Whence
$$
f'(m^2 + f(x)) f'(x) = 1.
$$
Thus $f'(x) = 1$.
So we obtain
$$
f(x) = a + x.
$$
Putting it back we get
$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1350094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Evaluating a function at a point where $x =$ matrix. Given $A=\left(
\begin{array} {lcr}
1 & -1\\
2 & 3
\end{array}
\right)$
and $f(x)=x^2-3x+3$ calculate $f(A)$.
I tried to consider the constant $3$ as $3$ times the identity matrix ($3I$) but the answer is wrong. Appreciate any ideas.
| $$f(A) = A^2 - 3A + 3I$$
But you know that $$A^2 = \begin{pmatrix}
-1 & -4 \\ 8 & 7
\end{pmatrix}$$
So, $$f(A) = \begin{pmatrix}
-1 & -4 \\ 8 & 7
\end{pmatrix} + \begin{pmatrix}
-3 & 3 \\ -6 & -9
\end{pmatrix} + \begin{pmatrix}
3 & 0 \\ 0 & 3
\end{pmatrix}$$
Simplifying leads to $$\bbox[10px, border: 2px solid... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Find 4 numbers which create a ratio Find four numbers which create ratio if its known that sum of first and last is equal to 14, sum of middle two is equal to 11 and sum of squares of all numbers is equal to 221
I got only that sum of product first with last and product of two middle is equal to 48
I have written this ... | If we interpret the phrase "four numbers which create ratio" as "four numbers that make a proportion", i.e. "the ratio of the first two numbers equals the ratio of the last two numbers," then we have four equations in the four variables $a,b,c,d$:
$$\frac ab=\frac cd$$
$$a+d=14$$
$$b+c=11$$
$$a^2+b^2+c^2+d^2=221$$
From... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1350394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
According to Stewart Calculus Early Transcendentals 5th Edition on page 140, in example 5, how does he simplify this problem? In Stewart's Calculus: Early Transcendentals 5th Edition on page 140, in example 5, how does
$$\lim\limits_{x \to \infty} \frac{\dfrac{1}{x}}{\dfrac{\sqrt{x^2 + 1} + x}{x}}$$
simplify to
$$\li... | Hint: Observe that, for $x>0$,
$$\large\sqrt{x^2+1\strut}=\sqrt{x^2(1+\tfrac{1}{x^2})}=\sqrt{x^2\strut} \;\cdot\;\sqrt{1+\tfrac{1}{x^2}}=x\;\cdot\;\sqrt{1+\tfrac{1}{x^2}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1350812",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Calculus - Solving limits with square roots I am having trouble understanding how to solve this limit by rationalizing. I have the problem correct (I used Wolfram Alpha of course), but I still don't understand how it is completed. I was trying to solve this by multiplying both the numerator and the denominator by $\sqr... | $$\begin{align}
\lim_{x\to 5}\frac{\sqrt{x^2+11}-6}{x-5}
&= \lim_{x\to 5}\frac{\left(\sqrt{x^2+11}-6\right)
\left(\sqrt{x^2+11}+6\right)}{(x-5)\left(\sqrt{x^2+11}+6\right)} \\[2ex]
&= \lim_{x\to 5}\frac{\left(\sqrt{x^2+11}\right)^2-6^2}
{(x-5)\left(\sqrt{x^2+11}+6\right)} \\[2ex]
&= \lim_{x\to 5}\frac{x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1353339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Hypergeometric distribution exercise! A store has $20$ guitars in stock but 3 are defective. Claire buys $5$ guitars from this lot.
(a) Find the probability that Claire bought $2$ defective guitars.
I use $N=20,n=5, k = 3,x=2$ where $N$ is the total sample space, $n$ is the number of trials, $k$ is the number of defect... | We assume that Claire's selection method was such that all collections of $5$ guitars were equally likely to be bought.
There are $\binom{20}{5}$ ways to choose $5$ guitars from $20$.
There are $\binom{3}{2}$ ways to choose $2$ defectives from $3$. For each of these ways, there are $\binom{17}{3}$ ways to choose $3$ no... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1354136",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Divisibility question Prove:
(A) sum of two squares of two odd integers cannot be a perfect square
(B) the product of four consecutive integers is $1$ less than a perfect square
For (A) I let the two odd integers be $2a + 1$ and $2b + 1$ for any integers $a$ and $b$. After completing the expansion for sum of their... | A)
\begin{align*}
&\,(2a+1)^2+(2b+1)^2=(4a^2+4a+1)+(4b^2+4b+1)=4(a^2+b^2)+4(a+b)+2\\
=&\,2[2(a^2+b^2)+2(a+b)+1].
\end{align*}
Since $2(a^2+b^2)$ and $2(a+b)$ are both even, the expression between the brackets is odd because of the $+1$ term. Now, the double of an odd number can never be a perfect square. (Try proving t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1354723",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Integrating the given function involving trigonometric functions Find $\int \csc^{p/3}x \sec^{q/3} x dx $
Given - $(p,q \in I^{+} )$ and $(p+q=12)$
I tried to substitute $q = 12-p$ in the integral but didn't find anything satisfactory.
| $$
\int \csc ^{p/3}(x) \sec^{q/3}(x)dx = \int \frac{1}{\sin^{p/3}(x)\cos^{4-\frac{p}{3}}(x)}dx = \int \frac{1}{\cos^4 x \cdot \tan^{p/3}{x}}dx = \int \frac{\sec^4 x}{\tan ^{p/3}x}dx
$$
Let $u=\tan x \implies du = \sec^2 x dx \implies$
$$
\int\frac{(1+u^2)du}{u^{p/3}} = \frac{3}{3-p}u^{\frac{3-p}{3}} + \frac{3}{9-p}u^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1358541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Integral which must be solved using integration by parts I have to solve this problem using integration by parts. I am new to integration by parts and was hoping someone can help me.
$$\int\frac{x^3}{(x^2+2)^2} dx$$
Here is what I have so far:
$$\int udv = uv-\int vdu $$
$$u=x^2+2$$ Therefore, $$xdx=\frac{du}{2}$$
$$dv... | Hint:
$\int\frac{x^3}{(x^2+2)^2}dx$
Write $x^3asx^2x$
$\int\frac{x^2x}{(x^2+2)^2}dx$
*
*add and substract 2 in numerator
$\int\frac{[(x^2+2)-2]x}{(x^2+2)^2}dx$
*
*separate it as two integrals
$\int\frac{(x^2+2)x}{(x^2+2)^2}dx-\int\frac{2x}{(x^2+2)^2}dx$
$=>\int\frac{x}{(x^2+2)}dx-\int\frac{2x}{(x^2+2)^2}dx$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1358927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
How prove $\lim x_n = \sqrt{\frac{a}{b}}$ if $b \neq 0$ for $x_1=c$ and $x_{n+1}=\frac{x_n^2 + \frac{a}{b}}{2x_n}$? Let $x_1=c$ and $x_{n+1}=\frac{x_n^2 + \frac{a}{b}}{2x_n}$. How prove $\lim x_n = \sqrt{\frac{a}{b}}$ if $b \neq 0$ ?
| For convenience, let $s:=\sqrt{\dfrac ab}$.
Then,
$$\frac{x_{n+1}-s}{x_{n+1}+s}=\frac{\dfrac{x_n^2+s^2}{2x_n}-s}{\dfrac{x_n^2+s^2}{2x_n}+s}=\left(\frac{x_n-s}{x_n+s}\right)^2,$$
and by recurrence
$$\frac{x_n-s}{x_n+s}=\left(\frac{c-s}{c+s}\right)^{2^n}.$$
This gives us the explicit formula
$$\color{green}{x_n=s\frac{1+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1359098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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Proving the integral series $\int _0^1\left(1-x^2\right)^n\,dx=\frac{2}{3}\cdot \frac{4}{5}\cdot\ldots\cdot \frac{2n}{2n+1}$
We have the series $\left(I_n\right)_{n\ge 1\:}$ where $$I_n=\int _0^1\left(1-x^2\right)^n\,dx.$$
Prove that $$I_n=\frac{2}{3}\cdot \frac{4}{5}\cdot\ldots\cdot \frac{2n}{2n+1}.$$
I tried to int... | As Byron Schmuland has pointed out a recursive pattern was developed here. The process in this solution will be a connection to the Beta function.
Consider the integral
\begin{align}
I_{n} = \int_{0}^{1} \left(1-x^{2}\right)^{n} \, dx.
\end{align}
Let $t = x^{2}$ to obtain
\begin{align}
I_{n} &= \frac{1}{2} \, \int_{0}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1359761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 5,
"answer_id": 2
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$a_1,a_2,...,a_n$ are positive real numbers, their product is equal to $1$, show: $\sum_{i=1}^n a_i^{\frac 1 i} \geq \frac{n+1}2$ it says to use the weighted AM-GM to solve it, because the inequality is not homogenous I've tried to use $$\lambda _ i = \frac{a_i^{\frac1i -1}}{\sum_{k=1}^n a_k^{\frac1k -1}}$$
this $\lamb... | \begin{align}
&a_1 + \sqrt{a_2} + \sqrt[3]{a_3} + ... + \sqrt[n]{a_n} \\
=& \sum_{k=1}^n \sum_{j=1}^k \frac{1}{k} \sqrt[k]{a_k}\\
\ge & \frac{n(n+1)}{2} \left(\prod_{k=1}^n(\frac{1}{k}\sqrt[k]{a_k})^k\right)^{\frac{2}{n(n+1)}} \\
= & \frac{n(n+1)}{2} \left(\prod_{k=1}^n(\frac{1}{k^k}{a_k})\right)^{\frac{2}{n(n+1)}} \\
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1360818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
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Computing a double gamma-digamma-trigamma series What are your thoughts on this series?
$$\sum _{k=1}^{\infty } \sum _{n=1}^{\infty } \frac{\Gamma (k)^2 \Gamma (n) }{\Gamma (2 k+n)}((\psi ^{(0)}(n)-\psi ^{(0)}(2 k+n)) (\psi ^{(0)}(k)-\psi ^{(0)}(2 k+n))-\psi ^{(1)}(2 k+n)).$$
EDIT: Noting the interest for this series, ... | I'd try to use: $$\displaystyle \int\limits_{0<x+y<1} x^{u-1}y^{u-1}(1-x-y)^{v-1} \,dx\,dy= \frac{\Gamma(u)^2\Gamma(v)}{\Gamma(2u+v)}$$
for reals $u,v \in \mathbb{R}$.
Hence, $$\displaystyle \begin{align}\sum\limits_{n,k=1}^{\infty} \frac{\Gamma(k+u)^2\Gamma(n+v)}{\Gamma(2k+2u+n+v)} &= \int\limits_{0<x+y<1} \frac{(xy)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1362440",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
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Prove that $\cos^2(\theta) + \cos^2(\theta +120^{\circ}) + \cos^2(\theta-120^{\circ})=3/2$
Prove that $$\cos^2(\theta) + \cos^2(\theta +120^{\circ}) + \cos^2(\theta-120^{\circ})=\frac{3}{2}$$
I thought of rewriting $$\cos^2(\theta +120^{\circ}) + \cos^2(\theta-120^{\circ})$$ as $$\cos^2(90^{\circ}+ (\theta +30^{\circ... | I am assuming that $\theta$ is in degrees. Then using the following formulas:
\begin{eqnarray}
\cos(x+y)&=&\cos x\cos y-\sin x\sin y\\
\cos(x-y)&=&\cos x\cos y+\sin x\sin y\\
(a-b)^2+(a+b)^2&=&2(a^2+b^2)\\
\cos(120)&=&-\frac12\\
\sin(120)&=&\frac{\sqrt3}{2}\\
\cos^2(x)+\sin^2(x)&=&1
\end{eqnarray}
we have:
\begin{eqna... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1364788",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 8,
"answer_id": 1
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Closed form for multiplicative recurrence relation In this StackOverflow question, I found an interesting recurrence relation:
$$f(n) = \begin{cases} 1 & n \leq 2 \\ nf(n-1) + (n-1)f(n-2) & \text{otherwise.}\end{cases}$$
I plugged it into Wolfram Alpha, and it gives me the solution:
$$f(n) = \frac{2~\Gamma(n+3) - 5~!(n... | Note that $\Gamma(n+3)=(n+2)!$.
$$f(n) = \frac{2\Gamma(n+3) - 5!(n+2)}{n+1}=\frac{2(n+2)! - 5 \cdot !(n+2)}{n+1}$$
Once one has this form, it is not hard using induction:
\begin{align*} f(n+1)&=(n+1)f(n)+nf(n-1) \\ &=\frac{2(n+2)! - 5 \cdot !(n+2)}{n+1}(n-2)+\frac{2(n+1)! - 5 \cdot !(n+1)}{n}n \\ &= 2(n+2)! - 5 \cdot !... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1367479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Definite Integration with Trigonometric Substitution I'm working on a question that involves using trigonometric substitution on a definite integral that will later use u substitution but I am not sure how to go ahead with this.
$$\int_1^2\frac1{x^2\sqrt{4x^2+9}}dx$$
My first step was to use $\sqrt{a^2+x^2}$ as $x=a\ta... | \begin{align*}\int\frac{\frac32\sec^2\theta\,\mathrm d\mkern1.5mu\theta}{\frac94\tan^2\theta\sqrt{9\tan^2\theta+9}}&=\frac29\int\frac{\mathrm d\mkern1.5mu\theta}{\sin^2\theta\sqrt{1+\tan^2\theta}}=\frac29\int\frac{\lvert\cos\theta\rvert\,\mathrm d\mkern1.5mu\theta}{\sin^2\theta}\\[1ex]
&=\frac29\int\frac{\cos\theta\,\m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1369251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Prove the inequality $\sqrt\frac{a}{a+8} + \sqrt\frac{b}{b+8} +\sqrt\frac{c}{c+8} \geq 1$ with the constraint $abc=1$ If $a,b,c$ are positive reals such that $abc=1$, then prove that $$\sqrt\frac{a}{a+8} + \sqrt\frac{b}{b+8} +\sqrt\frac{c}{c+8} \geq 1$$ I tried substituting $x/y,y/z,z/x$, but it didn't help(I got the r... | Let $a=\frac{x^2}{yz}$, $b=\frac{y^2}{xz}$ and $c=\frac{z^2}{xy}$, where $x$, $y$ and $z$ are positives.
Hence, by Holder and AM-GM we obtain:
$$\sum_{cyc}\sqrt{\frac{a}{a+8}}=\sum_{cyc}\frac{x}{\sqrt{x^2+8yz}}=\sqrt{\frac{\left(\sum\limits_{cyc}\frac{x}{\sqrt{x^2+8yz}}\right)^2\sum\limits_{cyc}x(x^2+8yz)}{\sum\limits_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1369441",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 6,
"answer_id": 1
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Formulae for sequences Given that for $1^3 + 2^3 + 3^3 + \cdots + n^3 = \frac{n^2(n+1)^2}{4}$
deduce that $(n+1)^3 + (n+2)^3 +\cdots+ (2n)^3 = \frac{n^2(3n+1)(5n+3)}{4}$
So far:
the sequence $(n+1)^3 + (n+2)^3 +\cdots+ (2n)^3$ gives $2^3 + 3^3 + 4^3 +\cdots,$ when n=1.
The brackets in the formula for the second sequenc... | Adding the two sums, you will find the sum for $2n$.
Indeed,
$$n^2(n+1)^2+n^2(3n+1)(5n+3)=n^2(16n^2+16n+4)=(2n)^2(2n+1)^2.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1370530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
if $(1-a)(1-b)(1-c)(1-d) = \frac{9}{16}$ then minimum integer value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} = ?$ Given $a,b,c,d > 0$, how do we find the minimum integer value of $n=\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d}$ such that $(1-a)(1-b)(1-c)(1-d) = \frac{9}{16}$.
| If $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\leq 1$, then $a,b,c,d>1$ and
$$\begin{align}
0\leq abcd-bcd-cda-dab-abc=&(1-a)(1-b)(1-c)(1-d)-1+(a+b+c+d)
\\&-(ab+ac+ad+bc+bd+cd)
\\
=&\frac{9}{16}-1-a(b-1)-b(c-1)-c(d-1)-d(a-1)
\\
&-ac-bd
\\
<&-\frac{7}{16}\,,
\end{align}
$$
which is a contradiction. Hence, $\frac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1372477",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Polynomial of $11^{th}$ degree Let $f(x)$ be a polynomial of degree $11$ such that $f(x)=\frac{1}{x+1}$,for $x=0,1,2,3.......,11$.Then what is the value of $f(12)?$
My attempt at this is:
Let $f(x)=a_0+a_1x+a_2x^2+a_3x^3+......+a_{11}x^{11}$
$f(0)=\frac{1}{0+1}=1=a_0$
$f(1)=\frac{1}{1+1}=\frac{1}{2}=a_0+a_1+a_2+a_3+...... | HINT:
Let $(x+1)f(x)=1+A\prod_{r=0}^{11}(x-r)$ where $A$ is an arbitrary constant
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1374128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 0
} |
Solve $10x^4-7x^2(x^2+x+1)+(x^2+x+1)^2=0$ How to solve the following equation?
$$10x^4-7x^2(x^2+x+1)+(x^2+x+1)^2=0$$
My attempt:
$$ 10x^4 - (7x^2+1)(x^2+x+1)=0$$
Thats all i can
Update
Tried to open brakets and simplify:
$$(7x^2+1)(x^2+x+1) = 7x^4+7x^3+7x^2+x^2+x+1=7x^4+7x^3+8x^2+1 $$
$$10x^4 - (7x^2+1)(x^2+x+1)= 3x^4... | Divide by $(x^2 + x + 1)^2$, the equation becomes: $$10\frac{x^4}{(x^2 + x + 1)^2} - 7\frac{x^2}{x^2 + x + 1} + 1 = 0$$ Let $z = \frac{x^2}{x^2 + x + 1}$. The equation now is $$10z^2 - 7z + 1 = 0$$ Solving it like an ordinary quadratic equation on $z$ you get at most two roots $z_{1,2}$. Then let $\frac{x^2}{x^2 + x + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1374321",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
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Am I getting the right answer for the integral $I_n= \int_0^1 \frac{x^n}{\sqrt {x^3+1}}\, dx$?
Let $I_n= \int_0^1 \dfrac{x^n}{\sqrt {x^3+1}}\, dx$. Show that $(2n-1)I_n+2(n-2)I_{n-3}=2 \sqrt 2$ for all $n \ge 3$. Then compute $I_8$.
I get an answer for $I_8={{2 \sqrt 2} \over 135}(25-16 \sqrt 2)$, could somebody plea... | \begin{align*}I_n&=\int_0^1 \dfrac{x^n}{\sqrt{x^3+1}}dx=\int_0^1 \dfrac{x^{n-3}(x^3+1-1)}{\sqrt{x^3+1}}dx = \int_0^1 x^{n-3}\sqrt{x^3+1}dx - \int_0^1 \dfrac{x^{n-3}}{\sqrt{x^3+1}}dx\\ &= \int_0^1 x^{n-3}\sqrt{x^3+1}dx-I_{n-3}\end{align*}This integral is handled with integration by parts:
$$\int_0^1 x^{n-3}\sqrt{x^3+1}d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1375090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
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Prove that $\tan\alpha =\tan^{2}\frac{A}{2}.\tan\frac{B-C}{2}$ Given a triangle ABC with the sides $AB < AC$ and $AM, AD$ respectively median and bisector of angle $A$. Let $\angle MAD = \alpha$. Prove that $$\tan\alpha =\tan^{2}\frac{A}{2}\cdot \tan\frac{B-C}{2}$$
| Firstly, in figure 1,
Using Napier's Analogy in $\Delta ABC$, we have,
$$\tan \left(\frac{B-C}{2}\right) = \frac{b-c}{b+c} \cdot \cot \left( \frac{A}{2}\right)$$
Now, in figure 2,
Using Sine Law in $\Delta ABM$, we have,
$$\dfrac{x}{\sin \left(\frac{A}{2} + \alpha \right)}=\dfrac{c}{ \sin(\angle BMA)}$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1375285",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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How to solve this equality? [3] $$4x^2 - 6x^4 + \frac{8x^6 - 2x^2 - \frac{1}{x^2}}{16} = 0$$
The equation has a strange look, and as such is probably as it should not be solved. Maybe the roots of trigonometric functions are expressed in terms of angles species $\frac{\pi}{n}$?
| $$4x^2-6x^4+\frac{8x^6-2x^2-\frac{1}{x^2}}{16}=0$$
$$4x^2-6x^4+\frac{\frac{8x^8-2x^4-1}{x^2}}{16}=0$$
$$4x^2-6x^4+\frac{8x^8-2x^4-1}{16x^2}=0$$
$$\frac{64x^4-96x^6+8x^8-2x^4-1}{16x^2}=0$$
$$8x^8-96x^6+64x^4-2x^4-1=0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1375590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Struggling with an inequality: $ \frac{1}{\sqrt[n]{1+m}} + \frac{1}{\sqrt[m]{1+n}} \ge 1 $ Prove that for every natural numbers, $m$ and $n$, this inequality holds:
$$
\frac{1}{\sqrt[n]{1+m}} + \frac{1}{\sqrt[m]{1+n}} \ge 1
$$
I tried to use Bernoulli's inequality, but I can't figure it out.
| The AM-GM inequality says that for $x,y\ge0$ and $0\le a\le1$,
$$
ax+(1-a)y\ge x^ay^{1-a}\tag{1}
$$
substituting $x\mapsto x/a$ and $y\mapsto y/(1-a)$ yields
$$
x+y\ge\frac{x^ay^{1-a}}{a^a(1-a)^{1-a}}\tag{2}
$$
Therefore, with $x=(m+1)^{-1/n}$, $y=(n+1)^{-1/m}$, and $a=\frac n{m+n}$, we get
$$
\begin{align}
(m+1)^{-1/n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1377289",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 1
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Prove using mathematical induction that $x^{2n} - y^{2n}$ is divisible by $x+y$ Prove using mathematical induction that
$(x^{2n} - y^{2n})$ is divisible by $(x+y)$.
Step 1: Proving that the equation is true for $n=1 $
$(x^{2\cdot 1} - y^{2\cdot 1})$ is divisible by $(x+y)$
Step 2: Taking $n=k$
$(x^{2k} - y^{2k})$ is... | Step 1: putting $n=1$, we get $$x^{2n}-y^{2n}=x^2-y^2=(x-y)(x+y)$$ above number is divisible by $(x+y)$. Hence the statement is true $n=1$
step 2: assuming that for $n=m$, $(x^{2n}-y^{2n})$ is divisible by $(x+y)$ then we have $$(x^{2m}-y^{2m})=k(x+y) \tag 1$$
Where, $k$ is an integer
step 3: putting $n=m+1$ we get... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1377927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 1
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How to integrate ${x^3}/(x^2+1)^{3/2}$? How to integrate $$\frac{x^3}{(x^2+1)^{3/2}}\ \text{?}$$
I tried substituting $x^2+1$ as t, but it's not working
| Alternative approach:
Let $x=\tan\theta$, $dx=\sec^2\theta d\theta$
\begin{align}
\int\frac{x^3dx}{(x^2+1)^{3/2}}&=\int\frac{\tan^3\theta\cdot\sec^2\theta d\theta}{\sec^3\theta}\\&=\int\frac{\sin^3\theta d\theta}{\cos^2\theta}\\&=-\int\frac{\sin^2\theta d\cos\theta}{\cos^2\theta}\\&=\int1-\sec^2\theta d\cos\theta\\&=\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1378025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
Is $\left(45+29\sqrt{2}\right)^{1/3} + \left(45-29\sqrt{2}\right)^{1/3}$ an integer? The problem is the following:
Prove that this number
$$x = \left(45+29\sqrt{2}\right)^{1/3} + \left(45-29\sqrt{2}\right)^{1/3}$$
is an integer. Show which integer it is.
I thought that it has some relations with something like c... | Let $u=\sqrt[3]{45+29\sqrt2}$ and $v=\sqrt[3]{45-29\sqrt2}$; so, $x=u+v$. We have
$$
u^3 + v^3=90,\\
uv = \sqrt[3]{45+29\sqrt2}\cdot \sqrt[3]{45-29\sqrt2} = \sqrt[3]{45^2-29^2\cdot2} = \sqrt[3]{343} = 7
$$
But
$$
u^3 + v^3 = (u+v)(u^2 - uv + v^2) = (u+v)((u+v)^2 - 3uv),
$$
and
$$
90 = x(x^2 - 21).
$$
Since $x$ must be ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1378183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Find all values that solve the equation For which values a, the equation
$$ a\sin{x}+(a+1)\sin^2{\frac{x}{2}} + (a-1)\cos^2{\frac{x}{2}} =1 $$
has a solution?
My idea: I think it's possible to factorize equation or reduce equation to the form like: $a(\sin^2{\frac{x}{2}} + \cos^2{\frac{x}{2}}) =1 $
Let's go:
$$ 2a\sin{... | $$ a\sin x + a(\sin^2 (x/2) + \cos^2 (x/2)) + \sin^2 (x/2) - \cos^2 (x/2) =1$$
$$ a\sin x + a(\sin^2 (x/2) + \cos^2 (x/2)) + 1 - 2\cos^2 (x/2) =1$$
$$ a \sin x + a - \cos x =1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1378266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Find the equation of the circle. Find the equation of the circle whose radius is $5$ which touches the circle $x^2 + y^2 - 2x -4y - 20 = 0$ externally at the point $(5,5)$
| HINT:
$(x-a)^2+(y-b)^2=5^2$ will touch $(x-1)^2+(y-2)^2=5^2$
iff $5+5=\sqrt{(a-1)^2+(b-2)^2}$
Again, $(a,b), (1,2), (5,5)$ are collinear.
So, we have two equations with two unknowns
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1378916",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
The value of the definite integral The value of the definite integral $\displaystyle\int\limits_0^\infty \frac{\ln x}{x^2+4} \, dx$ is
(A) $\dfrac{\pi \ln3}{2}$ (B) $\dfrac{\pi \ln2}{3}$ (C) $\dfrac{\pi \ln2}{4}$ (D) $\dfrac{\pi \ln4}{3}$
I tried using integration by parts,
\begin{align}
& \int_0^\infty \frac{\ln x... | $\bf{My\; Solution::}$ Let $\displaystyle I = \int_{0}^{\infty}\frac{\ln(x)}{x^2+4}dx\;,$ Now Let $x=2t\;,$ Then $dx = 2dt$
and Changing Limits, we get
$$\displaystyle I = \int_{0}^{\infty}\frac{\ln(2t)}{4t^2+4}\times 2dt = \frac{1}{2}\int_{0}^{\infty}\frac{\ln(2t)}{t^2+1}dt=\frac{1}{2}\int_{0}^{\infty}\frac{\ln(2)}{t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1378974",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Some help on trigonometric equation So I have $\sin^3x = \frac 34 \sin x$.
Can you expand so the answer is either $\sin x(\sin^2x +\frac 34)$ which leads to
the answer $\frac 12 + 2n\pi$ or that $\sin^3x = \frac 14(3\sin x-\sin^3x) - \frac 34\sin x$ which leads to the answer $0 + 2n \pi$.
Is that correct by any chance... | Notice, we have $$\sin^3x = \frac 34 \sin x$$ $$\sin^3x -\frac 34 \sin x=0$$ $$\sin x\left(\sin^2 x-\frac{3}{4}\right)=0$$ $$\text{if}\ \sin x=0\ \implies \color{blue}{x=n\pi}$$ $$\text{if}\ \sin^2 x-\frac{3}{4}\iff \sin^2x=\left(\frac{\sqrt{3}}{2}\right)^2 \iff \sin^2x=\left(\sin \frac{\pi}{3}\right)^2\ \implies \colo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1379971",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 5
} |
Matrix $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ to a large power Compute
$\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}^{99}$
What is the easier way to do this other than multiplying the entire thing out?
Thanks
| Consider the case of a general $2 \times 2$ matrix times $\left( \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right)$:
$\left( \begin{matrix} a & b \\ c & d \end{matrix} \right) \times \left( \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right) = \left( \begin{matrix} a & a + b \\ c & c + d \end{matrix} \right) $.
Thus, w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1380775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Show by induction that $F_n \geq 2^{0.5 \cdot n}$, for $n \geq 6$ I have the following problem:
Show by induction that $F_n \geq 2^{0.5 \cdot n}$, for $n \geq 6$
Where $F_n$ is the $nth$ Fibonacci number.
Proof
Basis
$n = 6$.
$F_6 = 8 \geq 2^{0.5 \cdot 6} = 2^{\frac{6}{2}} = 2^3 = 8$
Induction hypothesis
Assume $F... | We have that $F_n>F_{n-1}$ then $$F_{n+1}=F_n+F_{n-1}>2F_{n-1}>2\cdot2^{(n-1)/2}=2^{(n+1)/2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1381545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Subsets and Splits
Fractions in Questions and Answers
The query retrieves a sample of questions and answers containing the LaTeX fraction symbol, which provides basic filtering of mathematical content but limited analytical insight.