Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
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Closed form for a sum involving binomial coefficient $\sum_{j=0}^n \binom{n}{j} \frac1{j+1} = \frac{2^{n+1}-1}{n+1}$
Possible Duplicate:
How can I compute $\sum\limits_{k = 1}^n \frac{1} {k + 1}\binom{n}{k} $?
How to derive the following equality?
$$\sum_{j=0}^n \binom{n}{j} \frac1{j+1} = \frac{2^{n+1}-1}{n+1}$$
| Using the identity
$$
\binom{n}{j}\frac{n+1}{j+1}=\binom{n+1}{j+1}\tag{1}
$$
and summing
$$
\begin{align}
\sum_{j=0}^n\binom{n}{j}\frac{n+1}{j+1}
&=\sum_{j=0}^n\binom{n+1}{j+1}\\
&=2^{n+1}-1\tag{2}
\end{align}
$$
Dividing by $n+1$ yields
$$
\sum_{j=0}^n\binom{n}{j}\frac{1}{j+1}=\frac{2^{n+1}-1}{n+1}\tag{3}
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/87244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
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.
$$= (3-\... | This comes up pretty often, I think it is worth throwing in a bit more than you asked.
Suppose I have an $n$ by $n$ square matrix where all entries are equal to $1$:
$$ R_n \; \; = \; \;
\left( \begin{array}{ccccccc}
1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 ... | {
"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": 1
} |
Simplifying rational fractions I can't get this one either for whatever reason, spent about 20 minutes on it and I have made no progress at all.
$$\frac{x^2}{(x^2-4)} - \frac{x+1}{x+2}.$$
I know that I can simplify this into one fraction so I make it $$\frac{x^2}{x+2}-\frac{(x+1)(x^2-4)}{(x^2-4)(x+2)}$$
I then can simp... | $$\frac{x^2}{x^2-4} - \frac{x+1}{x+2} = \frac{x^2}{(x-2)(x+2)}-\frac{x+1}{x+2}=\frac{x^2}{(x-2)(x+2)}-\frac{(x-2)(x+1)}{(x-2)(x+2)}.$$
Working with the numerators:
$$
x^2-(x-2)(x+1) = x^2 - (x^2 -x -2).
$$
Here's the easiest mistake to make (I've seen this happen zillions of times including in calculus courses):
Righ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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How can I evaluate $\int_{-\infty}^{\infty}\frac{e^{-x^2}(2x^2-1)}{1+x^2}dx$? How can I solve this integral: $$\int_{-\infty}^{\infty}\frac{e^{-x^2}(2x^2-1)}{1+x^2}dx.$$ Can I solve this problem using the Laplace transform? How can I do this?
| To find $I=\int_{-\infty}^{\infty}\frac{e^{-x^2}(2x^2-1)}{1+x^2}dx$,
let us start by defining for $n \in \mathbb{N}$
$$
I_n = \int_{-\infty}^{\infty} x^{2n} e^{-x^2}\,dx
$$
and note that
$I_0=\sqrt{\pi}$, also called the Gaussian integral,
can be solved either by a beautiful polar coordinate trick
or by this technique,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/93741",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 5,
"answer_id": 0
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Determining $|z-1|$ when $z=\cos\theta +i\sin\theta$ and $\theta$ is acute As the question indicates we are supposed to find the modulus of z-1.
When trying to solve the problem I drew a diagram which you can see below:
The book I am working in solved a similar problem when showing examples, however the question was f... | If $z=x+iy$, then the modulus of $z$ is given by $\sqrt{x^2+y^2}$.
So with $z=\cos\theta +i\sin\theta$, we have $z-1=(\cos\theta-1)+i\sin\theta$. So the modulus is $$\sqrt{(\cos\theta-1)^2+\sin^2\theta}=\sqrt{\cos^2-2\cos\theta+1+\sin^2\theta}=\sqrt{2-2\cos\theta}$$
since $\cos^2\theta+\sin^2\theta=1$.
Using the identi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/94356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
solve complex equation $x^8 = \frac{1+i}{\sqrt{3} - i} = \frac{\sqrt[8]{\frac{2}{\sqrt{2}}}(\cos \frac{\pi}{4} + i \sin{\frac{\pi}{4}})}{2 \cos \frac{\pi}{6} + i \sin \frac{3\pi}{2}}$
What's the way to solve this kind of equation? I think there must be 8 solutions.
I tried to solve the following two equations
$a^6 = 1+... | HINT:
$$x^8=\frac{1+i}{\sqrt{3}-i}=\frac{1+i}{\sqrt{3}-i}.\frac{\sqrt{3}+i}{\sqrt{3}+i}=\frac{\sqrt{3}-1}{4}+i\frac{1-\sqrt{3}}{4}$$
Then use polar coordinates, $x^8=r(cos\theta +i\sin\theta)$ to obtain all the 8 roots.
Think of the identity $(r(cos\theta +i\sin\theta))^n=r^n(\cos n\theta+i\sin n\theta)$
(This is De-M... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/95226",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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How to solve the recurrence $T(n) = \frac{n}{2}T(\frac{n}{2}) + \log n$ I am trying to solve the recurrence below but I find myself stuck.
$T(n) = \frac{n}{2}T(\frac{n}{2}) + \log n$
I have tried coming up with a guess by drawing a recurrence tree. What I have found is
number of nodes at a level: $\frac{n}{2^{i}}$
ru... | The summation
$$ \sum\limits_{i=0}^n \frac{i}{2^{i}} $$
is equal to
$$ \sum\limits_{i=0}^n i(\frac{1}{2})^i $$
which is of the form
$$ \sum\limits_{i=0}^n ix^i, x = \frac{1}{2} $$
Consider the summation $$ \sum\limits_{i=0}^n x^i = \frac{x^{n+1} - 1}{x - 1} $$
By differentiating with respect to $x$ we have
$$ \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/102279",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Showing $24|(n+1)\implies24|\sigma(n)$ Question:
Show that if $n$ is a positive integer such that $24$ divides into $n + 1$, then $24$ divides the sum of all divisors of $n$ (denoted in number theory by $\sigma(n)$ or $\sigma_1(n)$).
For example if $n = 95$, then $n + 1 = 96 = 4 \times 24$ and the sum of the divisor... | We use a pairing argument, working first modulo $3$ and then modulo $8$.
We are told that $n\equiv -1\pmod{24}$. It follows that $n\equiv -1\pmod{3}$. Split the set of divisors of $n$ into unordered pairs $\{a,b\}$ such that $ab=n$. (Since $n\equiv -1\pmod 3$, the number $n$ is not a perfect square, so every divisor of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/103379",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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$(\cos \alpha, \sin \alpha)$ - possible value pairs We introduced the complex numbers as elements of $ \mathrm{Mat}(2\times 2, \mathbb{R})$ with
$$
\mathbb{C} \ni x =
\left(\begin{array}{cc}
a & -b \\
b & a \\
\end{array}\right) =
\frac{1}{\sqrt{a^2+b^2}}
\left(\begin{array}{cc}
\frac{a}{\sqrt{a^2+b^2}} & \frac... | There are two key factors here: one is the one noted, that each of these quantities is between $0$ and $1$.
But the other, which is very important, is that the sum of their squares is equal to $1$:
$$\left(\frac{a}{\sqrt{a^2+b^2}}\right)^2 + \left(\frac{b}{\sqrt{a^2+b^2}}\right)^2 = 1.$$
Because the sum of their square... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/105364",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Lim $x\to 0$ of $\frac{\sin(\pi x)}{\tan(\sqrt{3}x)}$ $$
\lim_{x \to 0} \frac{\sin(\pi x)}{\tan(\sqrt{3} x)}
$$
I need a step by step explanation. Thank you.
| Results going to be used:
*
*$\displaystyle \lim_{x \to 0} \frac{\sin{x}}{x} =1 = \lim_{x \to 0} \frac{\tan{x}}{x}$
What you have is
\begin{align*}
\lim_{x \to 0}\: \frac{\sin(\pi{x})}{\tan\sqrt{3}{x}} &= \lim_{x \to 0} \: \frac{\sin\pi{x}}{\pi{x}} \times \frac{\sqrt{3}x}{\tan{\sqrt{3}x}} \times \frac{\pi{x}}{\sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/106357",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Find $\lim\limits_{(x,y) \to (0,0)} \frac{e^{-\frac{1}{x^2+y^2}}}{x^4+y^4} $ I'm trying to find $$\lim\limits_{(x,y) \to (0,0)} \frac{e^{-\frac{1}{x^2+y^2}}}{x^4+y^4} .$$
After I tried couple of algebraic manipulation, I decided to use the polaric method.
I choose $x=r\cos \theta $ , $y=r\sin \theta$, and $r= \sqrt{x^... | Since $2(x^4+y^4)\geqslant(x^2+y^2)^2$, the ratio $r(x,y)=\dfrac{\mathrm e^{-1/(x^2+y^2)}}{x^4+y^4}$ is such that
$$
0\lt r(x,y)\leqslant u\left(\frac1{x^2+y^2}\right),
\quad\text{where}\ u:z\mapsto2z^2\mathrm e^{-z}.
$$
Now, $\frac1{x^2+y^2}\to+\infty$ when $(x,y)\to(0,0)$ and $u(z)\to0$ when $z\to+\infty$ (as the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/107171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Complex numbers equation: $z^4 = -16$ Probably dumb question, but I would like to ask it anyway.
I was to solve this equation:
$z^4 = -16$
At first glance, the way to solve it would be (as any other equation):
$z^2 = \sqrt{-1} * 4 \lor z^2 = -\sqrt{-1} * 4$
$z^2 = 4i \lor z^2 = -4i$
$z = 2\sqrt{i} \lor z = -2\sqrt{i} \... | We have
$$z^4 + 16 = (z^2 + 4)^2 - 8z^2 = (z^2 - 2\sqrt{2} z + 4)(z^2 + 2\sqrt{2} z + 4).$$
Anotherway $$z^2 - 2\sqrt{2} z + 4 = (z-\sqrt{2})^2 + 2 = (z-\sqrt{2})^2 -(\sqrt{2} i)^2 = (z-\sqrt{2} + \sqrt{2} i)(z-\sqrt{2} + \sqrt{2} i)).$$
The same for $z^2 + 2\sqrt{2} z + 4$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/109063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
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
$$\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{1}{a} + \sum\limits_{k = 1}^\infty {{{\left( { - 1} \righ... | A related identity is proven in this answer using residue theory. Here is a real approach to that identity.
Convergence of the Principal Value
We will look at the principal value
$$
\begin{align}
f(x)
&=\lim_{n\to\infty}\sum_{k=-n}^n\frac1{k+x}\tag{1a}\\
&=\frac1x-\sum_{k=1}^\infty\frac{2x}{k^2-x^2}\tag{1b}
\end{align... | {
"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": 0
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Product Chain Power Rule： Either it's the book or I am wrong. The problem: $x^3\sqrt{2x+4}$
$f(x):= x^3$, $g(x):= \sqrt{2x+4}$
$(f\times g)' = f^{\prime}g+fg^{\prime}$ thus it should be
$3x^2\sqrt{2x+4} + (x^3)[\frac{1}{2}(2x+4)^{\frac{-1}{2}}(2)]$
which is $3x^2\sqrt{2x+4}+\frac{x^3}{\sqrt{2x+4}}$
The book gi... | As my colleagues have astutely pointed out, the product rule states $(fg)^{\prime} = f^{\prime} g + f g^{\prime}$. Define $f(x)= x^3$ and $g(x)= \sqrt{2x+4}$. As $f^{\prime}(x) = 3 x^{2}$ and $g^{\prime}(x) = \frac{1}{\sqrt{2x+4}}$, the product rule gives
\begin{align}
(f(x)g(x))^{\prime} = f^{\prime}(x) g(x) + f(x) g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/111551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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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 $$\arcta... | Regularized the series:
$$ \begin{eqnarray}
\sum_{n=0}^m \frac{1}{(3n+1)(3n+2)} &=& \sum_{n=0}^m \left( \frac{1}{3n+1} - \frac{1}{3n+2} \right) = \sum_{n=0}^m \int_0^1 \left( x^{3n} - x^{3n+1} \right) \mathrm{d} x \\
&=& \int_0^1 \left( \frac{(1-x^{3m+3}) (1-x)}{1-x^3} \right) \mathrm{d} x =
\int_... | {
"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": 1
} |
If $f(x)f(y)=f(\sqrt{x^2+y^2})$ how to find $f(x)$ As we know, for the $$f(x)f(y)=f(x+y)$$ $f(x)=\mathrm e^{\alpha x}$ is a solution.
What about
$f(x)f(y)=f(\sqrt{x^2+y^2})$?
Does anybody know about the solution of the function equation?
I tried to find $f(x)$.
See my attempts below to find $f(x)$.
$$f(x)=a_0+a_1x+\... | The answer to this question is a well known result called Maxwell's theorem, after James Clerk Maxwell. This earlier question deals with it:
very elementary proof of Maxwell's theorem
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/115784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 4,
"answer_id": 3
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How do I find complex roots of a quartic polynomial using quadratic formula? For something like:
$$
z^4 + 8z^2 + 3
$$
how can I find all the complex roots using the quadratic equation, or is there a better method?
I tried a u substitution using $u = z^2$, but then when I applied the quadratic formula, I was getting r... | Presumably you got $u=z^2=-4\pm\sqrt{13}$. Now you need to solve for $z$.
Both of our values of $z^2$ are negative. The solutions of $z^2=-4+\sqrt{13}$ are
$z=\pm i \sqrt{4-\sqrt{13}}$, and the solutions of $z^2=-4-\sqrt{13}$ are $z=\pm i\sqrt{4+\sqrt{13}}$.
Remark: Things get somewhat more complicated when you want... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/116602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Where have I gone wrong? Contour integration $\int_{-a}^a {u\over 1+u+u^2} du$ as $a\to \infty$ I would like to integrate $\int_{-a}^a {u\over 1+u+u^2} du$ as $a\to \infty$.
So I thought I might use the residue theorem. In the complex plane, the singularities occur at $z=e^{\pm i2\pi\over 3}$. So if we close the contou... | You can use the usual integration:
$$\eqalign{
& \int\limits_{ - a}^a {\frac{u}{{{u^2} + u + 1}}du} = \frac{1}{2}\int\limits_{ - a}^a {\frac{{2u}}{{{u^2} + u + 1}}du} \cr
& = \frac{1}{2}\left\{ {\int\limits_{ - a}^a {\frac{{2u + 1}}{{{u^2} + u + 1}}du} - \int\limits_{ - a}^a {\frac{{du}}{{{u^2} + u + 1}}} }... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/116879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Proving $x^n - y^n = (x-y)(x^{n-1} + x^{n-2} y + ... + x y^{n-2} + y^{n-1})$ In Spivak's Calculus 3rd Edition, there is an exercise to prove the following:
$$x^n - y^n = (x-y)(x^{n-1} + x^{n-2} y + ... + x y^{n-2} + y^{n-1})$$
I can't seem to get the answer. Either I've gone wrong somewhere, I'm overlooking something, ... | Since $x^{n-1} + x^{n-2} y + \dots + x y^{n-2} + y^{n-1}$ is a geometric series with $n$ terms and a common factor of $y/x$, it equals
$$
\frac{x^{n-1}\left(1-\left(\frac{y}{x}\right)^n\right)}{1-\frac{y}{x}}=\frac{x^{n}\left(1-\left(\frac{y}{x}\right)^n\right)}{x-y}=\frac{x^n-y^n}{x-y} \, ;
$$
multiplying through by $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/117660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
"answer_count": 9,
"answer_id": 7
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Prove $\left \lfloor \frac{1}2 \left( 2+\sqrt3 \right) ^{2002} \right\rfloor \equiv -1 \pmod7 $ Prove
$$ \large \left\lfloor \frac{1}2 \left( 2+\sqrt3 \right) ^{2002} \right\rfloor \equiv -1 \pmod7 $$
So far my intuion only tells me that this has something to do with $(2+\sqrt3)(2-\sqrt3)=1$, but I don't even know wher... | $$(2+\sqrt 3)^2=7+2\sqrt3$$
Now, $$(7+4\sqrt3)^{2m+1}+(7-4\sqrt3)^{2m+1}$$
$$=2\left(7^{2m+1}+\binom {2m+1}27^{2m-1}\cdot4^2\cdot3+\binom{2m+1}47^{2m-3}\cdot4^4\cdot3^2+\cdots+\binom{2m+1}{2m}7\cdot4^{2m}\cdot3^m\right)$$ which is divisible by $2\cdot 7=14.$
Consequently, $$\frac12(7+4\sqrt3)^{2m+1}+\frac12(7-4\sqrt3)^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/118578",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Steps for solving this inequality: $ x \geq \frac{6}{x - 1} $ Solve:
$$
x \geq \frac{6}{x - 1}
$$
I figure that I can't really multiply by both sides by $(x - 1)$ since I'm not sure if it will be positive or negative. So I multiplied by $(x - 1)^2$ to get:
$$
x^3 - 2x^2 -5x \geq -6
$$
but wasn't sure how to proceed... | $$
x \ge \frac{6}{x-1}
$$
Move everything to one side:
$$
x-\frac{6}{x-1} \ge 0
$$
The common denominator is $x-1$, so we have:
$$
\frac{x(x-1)}{x-1} - \frac{6}{x-1} \ge 0
$$
Simplify:
$$
\frac{x^2-x-6}{x-1}\ge 0
$$
Factor:
$$
\frac{(x-3)(x+2)}{x-1} \ge 0
$$
This changes signs at $-2$, $1$, and $3$. It is po... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/118804",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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How to evaluate $\displaystyle\int {1\over (1+kx^2)^{3/2}}dx$ What change of variable should I use to integrate $$\displaystyle\int {1\over (1+kx^2)^{3/2}}dx$$
I know the answer is $$\displaystyle x\over \sqrt{kx^2+1}.$$ Maybe a trig or hyperbolic function?
| For clarity purposes
$$
\begin{align*}
\dfrac{1}{2\sqrt{k}}\int \dfrac{1}{t^{\frac{3}{2}} \sqrt{t-1}}dt &= \dfrac{1}{2\sqrt{k}}\int \dfrac{\sqrt{t}}{t^{2} \sqrt{t-1}}dt\\
&= \dfrac{1}{2\sqrt{k}}\int \dfrac{1}{t^{2} \sqrt{1-\frac{1}{t}}}dt \tag{A}
\end{align*}
$$
Now in this integral above substitute
$$ \sqrt{1-\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/119184",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "4",
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Find a three digit number ($\overline{xyz}$)? (Excuse me for my english: I'm spanish speaker)
"Find a three digit number $\overline{xyz}$ such that $x^2 +y^2 + z^2$ is equal to the number (xyz). "
I have this equation:
$$x^2 -100x +(y^2-10y+z^2-z)=0$$
And the discriminant (in $x$):
$$\Delta_x = -4y^2 +40y-4z^2+4z+100... | Since $x^2+y^2+z^2\leq 3\times 9^2 = 243$, we know that $x=0$, $x=1$, or $x=2$. But if $x=2$, then $x^2+y^2+z^2 \leq 2\times 9^2 + 4 = 166$, so we must definitely have $x=0$ or $x=1$.
But $x=1$ doesn't work: that would give us a maximum possible total of $1+9^2+9^2 = 163$; that would meant that $y\leq 6$; but then the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/119710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Show that $2^{3^k}+1$ is divisible by $3^k$ for all positive integers $k$ I attempted this by induction: Here is what I did
For $k=1, 2^{3^{1}}+1 = 2^{3}+1 = 9 = 3^{2}$ is divisible by $3^{1}$ , so the result is true for $k=1$
Now I assume the result to be true for $k=m$,
$2^{3^{m}}+1$ is divisible by $3^m$. To show t... | Note that $2^{3^{m+1}}+1 = \left(2^{3^m} \right)^3 + 1 = \left(2^{3^m} + 1 \right)\left(\left(2^{3^m} \right)^2 - 2^{3^m} + 1 \right)$. By induction hypothesis, $3^m$ divides $\left(2^{3^m} + 1 \right)$. All you need to show is that $3$ divides $\left(\left(2^{3^m} \right)^2 - 2^{3^m} + 1 \right)$.
Hint: $2^{3^m} \equi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/119890",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Sum of the reciprocal of sine squared I encountered an interesting identity when doing physics homework, that is,
$$ \sum_{n=1}^{N-1} \frac{1}{\sin^2 \dfrac{\pi n}{N} } = \frac{N^2-1}{3}. $$
How is this identity derived? Are there any more related identities?
| Here it is another approach, just for fun. We may notice that $\cos\frac{\pi}{n+1},\cos\frac{2\pi}{n+1},\ldots,\cos\frac{n\pi}{n+1}$ are the roots of the polynomial
$$ U_n(x) = \sin((n+1)\arccos x)/\sqrt{1-x^2} $$
hence
$$ S_n=\sum_{k=1}^{n-1}\frac{1}{\sin^2\frac{\pi k}{n}}=\frac{1}{2\pi i}\oint \frac{1}{1-z^2}\cdot\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/122836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
"answer_id": 2
} |
sequence $U_k=a_0U_{k-1}+a_1U_{k-2}+\cdots+a_{k-1}U_0$ Is there a general formula for $U_k$ defined by
$$U_k=a_0U_{k-1}+a_1U_{k-2}+\cdots+a_{k-1}U_0$$
where the $a_i$ are in arithmetic progression and $U_0=1$? Do there always exist $c,d$ such that $U_k\to cd^k$ as $k\to\infty$?
If $a_n=n+1$,
$U_k=F_{2k}$ where $F_n$ i... | The recurrence equation is written as
$$
U_k = \sum_{m=0}^{k-1} a_{k-1-m} U_m \tag{1}
$$
Let's form the generating function $f(x) = \sum_{k=0}^\infty x^k U_k$. Then, summing eq. 1, multiplied on both sides with $x^k$, from $k=1$ to $\infty$:
$$
f(x) - 1 = x \cdot f(x) \cdot \sum_{k=0}^\infty a_k x^k = x f(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/123604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Finding $\int \frac{1}{x}\sqrt{1+\frac{1}{x^4}}\,dx$ Finding $$\int \frac{1}{x}\sqrt{1+\frac{1}{x^4}}\,dx$$
I suppose I need integration by parts and trigo substitution
Let $u=\frac{1}{x} \Rightarrow du = -\frac{1}{x^2} dx$
Let $dv = \sqrt{1+(\frac{1}{x^2})^2}$, $\frac{1}{x^2} = \tan{\theta}$. Is my substitution OK?
S... | $$ \begin{aligned}
\int\frac{1}{x}\sqrt{1+\frac{1}{x^4}}\,\mathrm{d}x&=\int \frac{\sqrt{x^4 +1}}{x^3}\,\mathrm{d}x\\
&=\int\frac{x^4 + 1}{x^3\sqrt{x^4+1}}\,\mathrm{d}x\\
&=\int\frac{x}{\sqrt{\left(x^2\right)^2+1}}\,\mathrm{d}x + \int \frac{1}{x^5}\frac{1}{\sqrt{1 + 1/x^4}}\,\mathrm{d}x\\
&=\frac{1}{2}\ln\Big(x^2 + \sqr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/130394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Find The Number of Squares Consider the set of points $$S=\{(x,y):x,y\text{ are non-negative integers}\le n\}$$ Find the number of squares that can be formed with vertices belonging to $S$ and sides parallel to the axes.
| I will need some drawing help. For an $n$ that is small but not too small, put dots at the $(n+1)\times (n+1)$ gridpoints with coordinates $(x,y)$, where $0\le x,y\le n$. Something like $n=5$ is good enough.
Now draw the diagonals that go in the Northwest to Southeast direction. The diagonal closest to the origin ha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/132614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
How do you integrate $\int \frac{1}{a + \cos x} dx$? How do you integrate $\int \frac{1}{a + \cos x} dx$? Is it solvable by elementary methods? I was trying to do it while incorrectly solving a homework problem but I couldn't find the answer.
Thanks!
| Let $ y = \frac{x}{2}$.
$$\frac{1}{a + \cos 2y} = \frac{1}{a -1 + 2\cos ^2 y} = \frac{\sec^2 y}{(a-1)\sec^2 y + 2} = \frac{\sec^2 y}{a + 1 + (a-1)\tan^2 y} $$
Thus
$$\int \frac{1}{a + \cos x} \text{d}x = \int \frac{2}{a + \cos 2y} \text{d}y $$
$$ = \int \frac{ 2\sec^2 y}{ a + 1 + (a-1)\tan^2 y} \text{d} y$$
Now make t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/134577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 5,
"answer_id": 3
} |
Improper Integral Question: $ \int_0 ^ \infty\frac{x\log x}{(1+x^2)^2} dx $ I have to test the convergence of the integral :
$$ \int_0 ^ \infty\frac{x\log x}{(1+x^2)^2} dx $$
Please suggest. Also, have to show that the value of the integral is zero ?
| You want to find
$$ \int_0 ^ \infty\frac{x\log x}{(1+x^2)^2} dx $$
First, we find a primitive of the integrand, this is, we need
$$ \int\frac{x\log x}{(1+x^2)^2} dx =F(x)$$
Use integration by parts with $u = \log x$ and $dv = \dfrac{x}{(1+x^2)^2}$.
This makes $du = dx/x$ and $v=-\dfrac{1}{2(1+x^2)}$, from where
$$ \int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/138208",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 3
} |
Chain rule for $ y = (x^2 + x^3)^4$ I am trying to find the derivative of $ y = (x^2 + x^3)^4$
and it seems pretty simple I get
$4(x^2+x^3)^3 (2x+3x^2)$ This seems to be the proper answer to me but the book gets
$4x^7 (x+1)^3 (3x+2)$ and I have no idea how that happened, what process the author went through or why. My... | You did the differentiation correctly. The book's solution is the same as your answer, but in simplified form.
To obtain the book's answer from yours: factor $x^2$ from the $(x^2+x^3)$ term in your expression and apply the rule $(ab)^n=a^nb^n$, factor $x$ from the $(2x+3x^2)$ term, and finally combine the $x^6$ and $x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/141428",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
A trigonometric identity: $(\sin x)^{-2}+(\cos x)^{-2}=(\tan x+\cot x)^2$ I've been trying to prove it for a while, but can't seem to get anywhere.
$$\frac{1}{\sin^2\theta} + \frac{1}{\cos^2\theta} = (\tan \theta + \cot \theta)^2$$
Could someone please provide a valid proof?
I am not allowed to work on both sides of t... | Hint :
$$\sin^2 \theta=\frac{\tan^2 \theta}{1+\tan^2 \theta}$$
$$\cos^2 \theta=\frac{1}{1+\tan^2 \theta}$$
$$\tan \theta = \frac{1}{\cot \theta}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/142252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
Proof that $\int_{-1}^{1}\sqrt{1-x^2}dx=\frac{\pi}{2}$? In my textbook appears that $\displaystyle\int_{-1}^{1}\sqrt{1-x^2}dx=\frac{\pi}{2}$
But where does this equation come from?
| Not the "cleverest" method, like the above, - but works!
$$\int_{-1}^1 \sqrt{1-x^2} \ dx$$
To compute that integral, one may substitute $x=\sin{t}$, and get:
$$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sqrt{1-\sin^2{t}} \ d(\sin{t})=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos{t}\cos{t} \ dt=\int_{-\frac{\pi}{2}}^{\frac{\pi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/143287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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Prove $\gcd(a+b,a^2+b^2)$ is $1$ or $2$ if $\gcd(a,b) = 1$
Assuming that $\gcd(a,b) = 1$, prove that $\gcd(a+b,a^2+b^2) = 1$ or $2$.
I tried this problem and ended up with
$$d\mid 2a^2,\quad d\mid 2b^2$$
where $d = \gcd(a+b,a^2+b^2)$, but then I am stuck; by these two conclusions how can I conclude $d=1$ or $2$?
And... | Since $\text{gcd}(a,b) = 1$, we have that there exists $x,y \in \mathbb{Z}$ such that $$ax+by = 1$$ Hence, we have that $$(a+b)x + b(y-x) = 1$$ and $$a(x-y) + (a+b)y = 1$$
Squaring and adding the two equations, we get that $$(a+b)^2 x^2 + b^2(y-x)^2 + 2b(a+b)x(y-x) + (a+b)^2 y^2 + a^2(x-y)^2 + 2a(a+b)y(x-y) = 2$$
Rearr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/153125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 4,
"answer_id": 1
} |
Using conjugates to find a limit with a cubic root: $\lim_{h\to 0}\frac{\sqrt[3]{h+1}-1}{h}$ I currently have $$\lim_{h\to 0}\frac{\sqrt[3]{h+1}-1}{h}$$
Now, I know that when you have square root instead of a cubic root it's easy. You just multiply by the conjugate and simplify afterwards. If it were a sqrt I know that... | You can't conjugate cubic roots the same way you do square roots. You don't get the answer you think you get if you multiply $(\sqrt[3]{h+1}-1)(\sqrt[3]{h+1}+1)(\sqrt[3]{h+1}+1)$. Multiply it out and see that you don't get $h+1-1$:
$$\begin{align*}
\left(\sqrt[3]{h+1}-1\right)\left(\sqrt[3]{h+1}+1\right)\left(\sqrt[3]{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/153578",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
How to find the closed form formula for this recurrence relation $ x_{0} = 5 $
$ x_{n} = 2x_{n-1} + 9(5^{n-1})$
I have computed: $x_{0} = 5, x_{1} = 19, x_{2} = 83, x_{3} = 391, x_{4} = 1907$, but cannot see any pattern for the general $n^{th}$ term.
| If you're not familiar with the method that Phira explained, divide both sides by $2^n$:
$$
\dfrac{x_{n}}{2^n} = \dfrac{x_{n-1}}{2^{n-1}} + \dfrac{9}{2}\left(\dfrac{5}{2}\right)^{n-1}
$$
Call $\dfrac{x_{n}}{2^n} = s_n$:
$$
s_n = s_{n-1} + \dfrac{9}{2}\left(\dfrac{5}{2}\right)^{n-1}
$$
If we keep expanding $s_{n-1}$ in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/154667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 3
} |
Plotting a quadratic equation in the $\,xy\,$- plane My question is:
Represent the following set of points in the $\,xy\,$- plane:
$$\left\{ (x,y)\,\, |\,\, x^2 + y^2 - 2x - 2y + 1 = 0 \right\}$$
What i got: $\,\,(x-2)^2 + (y-2)^2 = 1\,\,$
I am not getting what to do next.
Any help to solve this question would be grea... | The first problem is that you carried out the algebra incorrectly. When you complete the square with $x^2-2x$ you should get $(x-1)^2-1$, which you can verify by multiplying it out. Similarly, $y^2-2y=(y-1)^2-1$. Thus, $$\begin{align*}x^2+y^2-2x-2y+1&=(x-1)^2-1+(y-1)^2-1+1\\
&=(x-1)^2+(y-1)^2-1\;,
\end{align*}$$
and th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/156246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
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Evaluating $\int \frac{x^3}{(x^2 + 1)^\frac{1}{3}}dx$ $$\int \frac{x^3}{(x^2 + 1)^\frac{1}{3}}dx$$
I am suppose to make a $u$ substitution and to make this a rational integral and then evaluate it from there but I have no idea how to do that. There aren't any good examples of this in the book and I can not find any us ... | Let $x^2 + 1 = t^3$. Hence, we get that $2x dx = 3t^2dt$ and $xdx = \dfrac{3t^2}{2} dt$. Hence, we have that
\begin{align}
I & = \int \dfrac{x^3}{(x^2+1)^{1/3}} dx = \int \dfrac{x^2 (xdx)}{(x^2+1)^{1/3}}\\
& = \int \dfrac{t^3-1}{t} \dfrac{3t^2}{2} dt = \dfrac32 \int(t^4-t) dt\\
& = \dfrac32 \left(\dfrac{t^5}{5} - \dfra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/156553",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Area of a surface of revolution of $y = \sqrt{4x+1}$ $y = \sqrt{4x+1}$ for $1 \leq x \leq 5$
I really have no idea what to do with this problem, I attempted something earlier which I will not type up because it took me two pages.
$$y = \sqrt{4x+1}$$
$$\int 2 \pi \sqrt{4x+1} \sqrt{1 + \frac{4}{1+4x}}dx$$
$$2 \pi \int \... | From the last step::
$$ \frac{\pi}{2} \int \sqrt u \frac{\sqrt{4 +u}}{\sqrt u} du = \frac \pi 2 \int \sqrt{4 + u} du $$
substituting $4 + u = p \implies du = dp \;\;$, we get
$$ = \frac \pi 2 \int \sqrt p dp = \frac \pi 2 \frac{p^{3/2}}{3/2} = \frac \pi 3 (4+u)^{3/2} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/157150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Positive definite quadratic form $\sum_{i=1}^n x_i^2 + \sum_{1\leq i < j \leq n} x_{i}x_j$ Is $\sum_{i=1}^n x_i^2 + \sum_{1\leq i < j \leq n} x_{i}x_j$ positive definite?
Approach:
The matrix of this quadratic form can be derived to be the following
$$M := \begin{pmatrix}
1 & \frac{1}{2} & \frac{1}{2} & \cdots & \frac... | We can more generally show that the $n\times n$ determinant of $$\left|\begin{array}{cccc}a & b & \cdots & b \\ c & a & \ddots & \vdots\\ \vdots & \ddots & \ddots & b \\ c & \cdots & c & a\end{array}\right|$$
is $\left\{\begin{array}{ll}\frac{1}{c-b}(c(a-b)^n-b(a-c)^n) & \text{ if }b\neq c \\ (a-b+nb)(a-b)^{n-1}& \t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/159506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 7,
"answer_id": 2
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How did Ramanujan get this result? We know Ramanujan got this result
$$\sqrt{1+2\sqrt{1+3\sqrt{1+\cdots }}}=3$$
and he used the formula
$$x+n+a=\sqrt{ax+{{(n+a)}^{2}}+x\sqrt{a(x+n)+{{(n+a)}^{2}}+(x+n)\sqrt{\cdots }}}$$
where $x=2,n=1,a=0$ ,we get the first result, but I don't know how to prove it, can you help me?
| Just use the following formula repeatedly
$ n^2=1+(n-1) (n+1) $
then we have
$ 2=\sqrt{1+1\times 3} $
$ 2=\sqrt{1 +1 \sqrt{1+2\times 4}}$
$ 2=\sqrt{1 +1 \sqrt{1+2 \sqrt{1+3\times 5}}} $
$ 2=\sqrt{1 +1 \sqrt{1+ 2 \sqrt{1+ 3 \sqrt{1+4\times 6}}}} $
$ 2=\sqrt{1+1 \sqrt{1+2 \sqrt{1+ 3 \sqrt{1+4 \sqrt{1+5\times 7}}}}}$
$ 2=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/160771",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 2,
"answer_id": 1
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prove that $\frac{b+c}{a^2}+\frac{c+a}{b^2}+\frac{a+b}{c^2}\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ Assume: $a,b,c >0$ prove that :
$$\frac{b+c}{a^2}+\frac{c+a}{b^2}+\frac{a+b}{c^2}\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$$
| $$\sum_{cyc}\left(\frac{b+c}{a^2}-\frac{1}{a}\right)>\sum_{cyc}\left(\frac{b+c}{a^2}-\frac{2}{a}\right)=$$
$$=\sum_{cyc}\frac{c-a-(a-b)}{a^2}=\sum_{cyc}(a-b)\left(\frac{1}{b^2}-\frac{1}{a^2}\right)\sum_{cyc}\frac{(a-b)^2(a+b)}{a^2b^2}\geq0$$
Done!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/161318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
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Computing the derivative from the definition Using the limit definition of the derivative which I know is:
$$f'(x)=\lim_{h\to0}\left(\frac{f(x+h)-f(x)}{h}\right)$$
I am trying to solve this problem
$$f(x)= \frac{x}{x+2} $$
How do I go about properly solving this, I seemed to get
$$\frac{x}{x+2}\ $$
as my answ... | I think that you mean that you want to use the definition of the derivative to find the derivative of the function $$f(x)=\frac{x}{x+2}\;.$$ The definition is that $$f\,'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}h\;.$$ Note the limit: it’s essential.
For your function this is
$$\begin{align*}
\lim_{h\to 0}\frac{\frac{x+h}{(x+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/161741",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
Slope of curve in $\mathbb{R}^3$ While doing revision, I came across this problem:
The surface given by $z=x^2-y^2$ is cut by the plane given by $y=3x$, producing a curve in the plane. Find the slope of this curve at the point $(1,3,-8)$.
I tried substituting $y=3x$ into $z=x^2-y^2$, yielding $z=-8x^2$. Then, $\frac{dz... | This is a very badly posed question, and does not have an answer. (Read the comments.)
The following is a solution to a rephrased question which can be answered, however.
Question:
The surface given by $z=x^2−y^2$ is cut by the plane given by $y=3x$, producing a curve in the plane.
Treating the intersection as a curve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/161996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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Integration Example How can i find the integration of this example
$$\int \frac{\sin x}{\sin x - \cos x } dx$$
I tried first add cos and then substracting cos but then what about $$\int \frac{\cos x}{\sin x - \cos x } dx\ ?$$
| Now let
$$ I = \int \frac{\cos x}{\cos x + \sin x} \, \mathrm{d}x \quad \text{and} \quad J = \int \frac{\sin x}{\cos x + \sin x} \, \mathrm{d}x$$
Notice that
\begin{align*}
I + J & = \int \frac{\cos x + \sin x}{\cos x + \sin x} \, \mathrm{d}x = x + \mathcal{C} \\
I - J & = \int \frac{(\sin x + \cos x)'}{\cos x + \sin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/162783",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Integral of determinant Good evening. I need help with this task
$$
\int\limits_{-\pi}^\pi\int\limits_{-\pi}^\pi\int\limits_{-\pi}^\pi{\det}^2\begin{Vmatrix}\sin \alpha x&\sin \alpha y&\sin \alpha z\\\sin \beta x&\sin \beta y&\sin \beta z\\\sin \gamma x&\sin \gamma y&\sin \gamma z\end{Vmatrix} \text{d}x\,\text{d}y\,\te... | Exact solution and Mathematica code to produce it (< 10 seconds computing time):
M = {{Sin[a x], Sin[a y], Sin[a z]},
{Sin[b x], Sin[b y], Sin[b z]},
{Sin[c x], Sin[c y], Sin[c z]}};
Expand[Det[M]^2] /. Plus -> List;
Total[Integrate[%, {x, -Pi, Pi}, {y, -Pi, Pi}, {z, -Pi, Pi}]]
It looks terrible but for a co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/163574",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
"answer_id": 2
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If $n\ne 4$ is composite, then $n$ divides $(n-1)!$. I have a proof and need some feedback. It seems really obvious that the statement is true but it is always the obvious ones that are a little trickier to prove. So I would appreciate any feedback. Thank you!
Here is what I am asked to prove:
If $n$ is composite then... | We will assume that $n>4$, since $4\hspace{-3pt}\not|\,3!$.
Let $p$ be the smallest factor of $n$. Since $n$ is composite, $p\le\sqrt{n}$.
If $p=\sqrt{n}$, then since $n>4$, we must have $p>2$ so that $2p<p^2=n$. Thus, $p\le n-1$ and $2p\le n-1$, and therefore, $2n=p\cdot2p\,|\,(n-1)!$
If $p<\sqrt{n}$, then $n/p>\sqrt{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/164852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "33",
"answer_count": 5,
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Given that $\tan^{-1}(x)+\tan^{-1}(y)+\tan^{-1}(xy)=11/12π$, prove that when $x=1, dy/dx=-1-\sqrt{3}/2$ Given that $x$ and $y$ satisfy the equation:
$$\arctan(x)+\arctan(y)+\arctan(xy)=11/12π$$
Prove that, when $x=1, dy/dx=-1-\sqrt{3}/2$.
I tried to differentiate both sides:
$$1/(1+x^2)+y/(1+y^2)+(y+x\,dy/dx)/(1+(xy)^2... | $$
\begin{align}
& \arctan x + \arctan y + \arctan(xy) = \arctan\left( \frac{x+y}{1-xy} \right) + \arctan(xy) \\
& = \arctan\left( \frac{\frac{x+y}{1-xy} + xy}{1-\left(\frac{x+y}{1-xy}\right)xy} \right) = \arctan\left( \frac{ x + y + xy - x^2 y^2}{1-xy -x^2 y - xy^2} \right) = \frac{11}{12} \pi.
\end{align}
$$
So
$$
\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/167784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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Show that $\left(1+\dfrac{1}{n}\right)^n$ is monotonically increasing
Show that $U_n:=\left(1+\dfrac{1}{n}\right)^n$, $n\in\Bbb N$, defines a monotonically increasing sequence.
I must show that $U_{n+1}-U_n\geq0$, i.e. $$\left(1+\dfrac{1}{n+1}\right)^{n+1}-\left(1+\dfrac{1}{n}\right)^n\geq0.$$
I am trying to go ahead... | Here's a proof I learned by solving problem 90 from Section 11.1 of James Stewart's Calculus: Early Transcendentals (8th edition). We'll need the following result:
Lemma: If $0\leq a<b$ and $n$ is a positive integer, then
$$\frac{b^{n+1}-a^{n+1}}{b-a}<(n+1)b^n$$
Proof: Define the function $f:[0,\infty)\rightarrow\math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/167843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "38",
"answer_count": 14,
"answer_id": 13
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How many $3\times 3$ binary matrices $X$ are there with determinant $0$ and $X^2=X^T$? How many $3 \times 3$ binary matrices $X$ are there with determinant as $0$ that also satisfy $X^2 = X^T$?
| There are $2^9=512$ binary $3\times 3$ matrices. Of these, $7\times 6\times 4 = 168$ are invertible, so there are $344$ singular ones.
Here's a somewhat naive way of going about it; I suspect there must be a clever/elegant way of doing it, but I couldn't think of one and then started going down this path. All computati... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove $\frac{\sec{A}+\csc{A}}{\tan{A} + \cot{A}} = \sin{A} + \cos{A}$ and $\cot{A} + \frac{\sin{A}}{1 + \cos{A}} = \csc{A}$ Can anyone help me solve the following trig equations.
$$\frac{\sec{A}+\csc{A}}{\tan{A} + \cot{A}} = \sin{A} + \cos{A}$$
My work thus far
$$\frac{\frac{1}{\cos{A}}+\frac{1}{\sin{A}}}{\frac{\sin{A... | Solution 1:
$$\dfrac{\dfrac{\sin{A} + \cos{A}}{\sin{A} \cos{A}}}{\dfrac{\sin^2{A} + \cos^2{A}}{\sin{A} \cos{A}}}$$
$$ = \frac{\sin{A} + \cos{A}}{\sin^2{A} + \cos^2{A}}$$
$$ = \sin{A} + \cos{A}$$
Solution 2:
$$\frac{\cos{A}(1 + \cos{A}) + \sin^2{A}}{\sin{A} (1 + \cos{A})}$$
$$= \frac{\color{red}{\cos{A} + 1}}{\sin{A}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/171675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
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Find an equation to transform $\{1, 2, 3, 4, 5,6, 7, 8\}$ to $\{8,7,6,5,4,3,2,1\}$ Is there a way to transform $\{1, 2, 3, 4, 5,6, 7, 8\}$ to $\{8,7,6,5,4,3,2,1\}$ by applying a single function?
That is, transform $1$ to $8$, $2$ to $7$, $3$ to $6$ etc.
| $f(x) = 9 - x$
$f(1) = 9 - 1= 8$
$f(3) = 9 - 3 = 6$
etc.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/172035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that the curve $y^2 = x^3 + 2x^2$ has a double point, and find all rational points Show that the curve $y^2 = x^3 + 2x^2$ has a double point. Find all rational points on this curve.
By implicit differentiation of $x$, $-3x^2 - 4x$ vanishes iff $x = -4/3$ and $0$.
By implicit differentiation of $y$, $2y$ vanishes i... | Since you are a visual person, you should try drawing it. It seems fairly obvious from the real picture that there is a double point at zero, and then you just use whatever methods your book usually employs to prove that it actually is a double point.
On the other hand, finding all rational solutions means finding all... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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The minimum value of $(\frac{1}{x}-1)(\frac{1}{y}-1)(\frac{1}{z}-1)$ if $x+y+z=1$ $x, y, z$ are three distinct positive reals such that $x+y+z=1$, then the minimum possible value of $(\frac{1}{x}-1) (\frac{1}{y}-1) (\frac{1}{z}-1)$ is ?
The options are: $1,4,8$ or $16$
Approach: $$\begin{align*}
\left(\frac{1}{x} -1\... | If you look at the statement of the AM-HM inequality correctly, you'd see that
$$
\frac 1x + \frac 1y + \frac 1z \ge 9
$$
and equality only happens when $x=y=z$. Therefore we can assume equality happens to find the minimum value, but $x+y+z = 1$ implies $x=y=z=1/3$. Therefore the minimum is $8$ and is attained uniquel... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/173492",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Factorise the determinant $\det\Bigl(\begin{smallmatrix} a^3+a^2 & a & 1 \\ b^3+b^2 & b & 1 \\ c^3+c^2 & c &1\end{smallmatrix}\Bigr)$ Factorise the determinant $\det\begin{pmatrix} a^3+a^2 & a & 1 \\ b^3+b^2 & b & 1 \\ c^3+c^2 & c &1\end{pmatrix}$.
My textbook only provides two simple examples.
Really have no idea how ... | $$\begin{align*}
\det\begin{pmatrix} a^3+a^2 & a & 1 \\ b^3+b^2 & b & 1 \\ c^3+c^2 & c &1\end{pmatrix} & = \det\begin{pmatrix} a^2 & a & 1 \\ b^2 & b & 1 \\ c^2 & c &1\end{pmatrix}+\det\begin{pmatrix} a^3 & a & 1 \\ b^3 & b & 1 \\ c^3 & c &1\end{pmatrix}\\
& = -(a-b)(b-c)(c-a) -(a-b)(b-c)(c-a)(a+b+c)\\
&= (a-b)(b-c)(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/173562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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Evaluating $\int ^\frac{\pi}{2}_{0} \sin\left(2x+\frac{\pi}{4}\right)\ dx$
Find the exact value of the following definite integral:
$$\int ^\frac{\pi}{2}_{0} \sin\left(2x+\frac{\pi}{4}\right)\:dx=\left[-\frac{1}{2}(2x+\frac{\pi}{4})\right]^\frac{\pi}{2}_{0}$$
$$=-\frac{1}{2}\left(2\frac{\pi}{2}+\frac{\pi}{4}\rig... | $$\int_0^{\pi/2}\sin(2x+\tfrac{\pi}{4})\,dx=[−\tfrac{1}{2}\cos(2x+\tfrac{π}{4})]_0^{\pi/2}=2\sqrt{2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/175815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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hints on solving DE How to solve this DE?
$$ {dx \over x} = {dy \over y} = {dz \over z - a \sqrt{x^2+y^2+z^2}}$$
From the first part, I get $y = c_1x$. How to find the other solution?
The answer according to answer sheet is $ z + \sqrt{x^2 + y^2 + z^2} = c_2$. Thank you for help.
| $$ {dx \over x} = {dy \over y} = {dz \over z - a \sqrt{x^2+y^2+z^2}}$$ You get $y=c_1x$, so put it into the third fraction:
$$ {dx \over x} = {dz \over z - a \sqrt{x^2+c_1^2x^2+z^2}}$$ $$ {dx \over x} = {dz \over z - a \sqrt{(1+c_1^2)x^2+z^2}}={dz \over z - a \sqrt{Cx^2+z^2}}$$ which is homogeneous equation: $$(z - a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/177980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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proof of a finite sum involving a binomial coefficient and a variable. I found that the following equation holds for integers $l$, $k$, and any $x \neq 0,1$,
$$\tag{1}
\sum\limits_{l = 0}^k {\left( { - 1} \right)^l } \left( {\begin{array}{*{20}c}
k \\
l \\
\end{array}} \right)\frac{{x^{l} }}{{\left( {\frac{{1 +... | We have, $$\frac{1}{\left(\frac lx + k - l+1 \right) \left(\frac{l+1}{x} + k-l \right)} = \frac{x}{1-x} \left(\frac{1}{\frac lx + k - l+1} - \frac{1}{\frac{l+1}{x} + k-l} \right)$$
Therefore, $$\sum_{l=0}^{k} (-1)^l \binom{k}{l} \frac{x^l}{\left(\frac lx + k - l+1 \right) \left(\frac{l+1}{x} + k-l \right)} = \frac{x}{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/178299",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How many unique distances are there in a 5 x 5 grid? I cannot figure this out:
I have a square in the plane with side length $5$. $A$ and $B$ are points in the square. The coordinates of $A$ and $B$ are always integers.
I want to know how many unique Euclidean distances are possible between $A$ and $B$.
I thought $15$... | If you count vertical or horizontal distances the values are 0,1,2,3,4,5;
If the horizontal (or vertical) has length 1 and the vertical has length 1, 2, 3, 4, 5 then the distances are $\sqrt{1^2+1^2}=\sqrt{2}, \sqrt{1^2+2^2}=\sqrt{5}, \sqrt{1^2+3^2}=\sqrt{10}, \sqrt{1^2+4^2}=\sqrt{17}, \sqrt{1^2+5^2}=\sqrt{26}$,
If t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/181016",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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For $A \in \mathbb{R}^{3 \times 3}$, find $P, Q \in \mathbb{R}^{3 \times 3}$ such that $A = P-Q$, where $P^2 = P$, $Q^2 = Q$ and $PQ = 0 = QP$ This is an exercise from a previous linear algebra exam:
The diagonalisable matrix $$A = \begin{pmatrix} 3 & -6 & 2\\ 4 & -7 & 2\\ 8 & -12 & 3 \end{pmatrix} \in \mathbb{R}^{3 \... | There's an invertible matrix $B$ such that $A=BDB^{-1}$, where $D$ is diagonal and has diagonal entries $\pm1$. $D=E-F$ where $E,F$ are diagonal, $E$ has 1 where $D$ does and 0 elsewhere, $F$ has 1 where $D$ has $-1$ and 0 elsewhere. Let $P=BEB^{-1}$, $Q=BFB^{-1}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/181636",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Probability without replacement question I have a bag full of 10 marbles: 2 black, 1 blue, 1 yellow, 3 green, 1 brown, and 2 purple. I draw 5 marbles one at a time without replacement. What is the probability of a black marble to be in the five drawn?
| At least one black: It is easier to find first the probability of no black.
There are $\binom{10}{5}$ ways to choose $5$ marbles, all equally likely. Note that there are $\binom{8}{5}$ ways to choose $5$ marbles from the $8$ non-black. So the probability that all the balls are non-black is
$$\frac{\binom{8}{5}}{\binom... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/182291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to evaluate $\lim\limits_{n\to+\infty} \prod\limits_{k=1}^n (1+k/n^2)$? I've got a limit which puzzle me several days. The question is
$$ \lim_{n\to+\infty} \prod_{k=1}^n\left(1+\frac{k}{n^2}\right).$$
Can you help me? Thank you in advance
| Intuitively, we have
$$\log\left( 1 + \frac{k}{n^2} \right) = \frac{k}{n^2} + O\left(\frac{1}{n^2}\right) \quad \Longrightarrow \quad \log \prod_{k=1}^{n} \left( 1 + \frac{k}{n^2} \right) = \frac{1}{2} + O\left(\frac{1}{n}\right)$$
and therefore the log-limit is $\frac{1}{2}$.
Here is a more elementary approach: Let $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/183061",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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How to find $x$ in some trigonometric equations How to solve these trigonometric equations?
$$\tan2x-\sin4x = 0$$
and
$$\tan2x = \sin x$$
I can't do this, please help me! I did this:
\begin{align}
\tan2x &= 2\sin x\\
\\
\frac{\sin2x}{\cos2x} &= \tan x
\end{align}
| $\tan2x=\sin4x$
$=>\frac{\sin2x}{\cos2x}=2\sin2x \cos2x$
$=>\sin2x(2cos^22x-1)=0$
$=>\sin2x\cos4x=0$
$\sin2x=0=>2x=n\pi=>x=\frac{r\pi}{2}$ for some integer r.
$\cos4x=0=>4x=\frac{(2r+1)\pi}{2}=>x=\frac{(2r+1)\pi}{8}$ for some integer r.
Again, $\sin4x=\sin x$,
$4x=n\pi+(-1)^nx$ for some integer n.
If $n$ is even $=2m(s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/183619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Calculating the residue of power towers I want to calculate the residue of a power tower. How do I do that?
For example, I want to know the answer to this:
$$2 \uparrow\uparrow 10 \pmod{10^9}$$
| When dealing with power towers with bases not relatively prime to the modulus, it's useful to employ the Chinese Remainder Theorem. And then repeatedly apply the Euler's Theorem.
$2 \uparrow \uparrow 10 \pmod {2^9} = 0$, so we only need to calculate $2 \uparrow \uparrow 10 \pmod{5^9}$.
By Euler's Theorem, we need to fi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/183910",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Finite summation
Possible Duplicate:
why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$
What is the proof without induction for :
$(1)$ $\sum_{i=1}^n\ i^2= \frac{n(n+1)(2n+1)}{6}$
$(2)$ $\sum_{i=1}^n\ i^3=\frac{n^2(n+1)^2}{4}$
| Let $S(n,t)=\sum_{1 ≤r ≤n} r^t$
We know,
$(r+1)^2-r^2=2\cdot r+1$
Putting r=1,2,3,...,n-1,n we get,
$(1+1)^2-1^2=2\cdot1+1$
$(2+1)^2-2^2=2\cdot2+1$
...
$(n-1+1)^2-(n-1)^2=2\cdot (n-1)+1$
$(n+1)^2-n^2=2\cdot n+1$
Adding both sides, $(n+1)^2-1=2S(n,1)+n$
=>$S(n,1)=\frac{n(n+1)}{2}$
We know, $(r+1)^3-r^3=3\cdot r^2+3\cdot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/184194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_count": 3,
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Proving that $\frac{\csc\theta}{\cot\theta}-\frac{\cot\theta}{\csc\theta}=\tan\theta\sin\theta$
Prove $\dfrac{\csc\theta}{\cot\theta}-\dfrac{\cot\theta}{\csc\theta}=\tan\theta\sin\theta$
So, LS=
$$\dfrac{\csc\theta}{\cot\theta}-\dfrac{\cot\theta}{\csc\theta}$$
$$\left(\dfrac{1}{\sin\theta}\cdot \dfrac{\tan\theta}{1}\... | $$\frac{\csc\theta}{\cot\theta}-\frac{\cot\theta}{\csc\theta}=\frac{\csc^2\theta-\cot^2\theta}{\cot\theta.\csc\theta}=\frac{1}{\cot\theta\csc\theta}=\frac{1}{\cot\theta}.\frac{1}{\csc\theta}=\tan\theta.\sin\theta$$
I have used the identity $\csc^2\theta-\cot^2\theta=1$ here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/185205",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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How to solve $100x +19 =0 \pmod{23}$ How to solve $100x +19 =0 \pmod{23}$, which is $100x=-19 \pmod{23}$ ?
In general I want to know how to solve $ax=b \pmod{c}$.
| $100x +19 =0 \pmod{23}$
$100x ≡ -19 \pmod{23}$
$92x+8x=4-23\pmod{23}$
$=>8x≡4 \pmod{23}$ dividing both sides by $23$ and taking resdiues,
So, $2x≡1 \pmod{23}$ dividing both sides by $4$ which is possible as $(4,23)=1,$
Now $\frac{23}{2}=11+\frac{1}{2}$ So, $23-2\cdot 11=1$
(Please refer to this and this, for the t... | {
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"url": "https://math.stackexchange.com/questions/186674",
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"question_score": "5",
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Proving $n^4 + 4 n^2 + 11$ is $16k\,$ for odd $n$
if $n$ is an odd integer, prove that $n^4 + 4 n^2 + 11$ is of the form
$16 k$.
And I went something like:
$$\begin{align*}
n^4 +4 n^2 +11
&= n^4 + 4 n^2 + 16 -5 \\
&= ( n^4 +4 n^2 -5) + 16 \\
&= ( n^2 +5 ) ( n^2-1) +16
\end{align*}$$
So, now we have to prove that the... | If $m$ is odd, then $m^2 \equiv 1$ (mod 8), since $(2a+1)^{2} = 4(a^{2}+a) +1$ and $a^2 +a$ is always even when $a$ is an integer. If $h$ is an integer congruent to $3$ (mod 8), then $h^{2}-9 = (h-3)(h+3)$ is divisible by $16$. Now when $n$ is odd, we have $n^{2}+2 \equiv 3$ (mod $8$), so $(n^{2}+2)^{2} \equiv 9$ (mod ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/187033",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "4",
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"answer_id": 3
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I know that, $S_{2n}+4S_{n}=n(2n+1)^2$. Is there a way to find $S_{2n}$ or $S_{n}$ by some mathematical process with just this one expression? $S_{2n}+4S_{n}=n(2n+1)^2$, where $S_{2n}$ is the Sum of the squares of the first $2n$ natural numbers, $S_{n}$ is the Sum of the squares of the first $n$ natural numbers.
when, ... | With the additional information you provided about $S_n$ (the sum of squares of the first $n$ integers), there's a neat solution (among other solutions) that uses the perturbation method described in Concrete Mathematics.
Let $C_n$ denote the sum of cubes of the first $n$ natural numbers. Then
\begin{equation}
\begin{s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Deriving the addition formula of $\sin u$ from a total differential equation How do we derive the addition formula of $\sin u$ from the following equation?
$$\frac{dx}{\sqrt{1 - x^2}} + \frac{dy}{\sqrt{1 - y^2}} = 0$$
Motivation
Let $u = \int_{0}^{x}\frac{dt}{\sqrt{1 - t^2}}$
Then $x = \sin u$
Let $v = \int_{0}^{y}\fra... | Let $u = \int_{0}^{x}\frac{dt}{\sqrt{1 - t^2}}$.
Then $x = \sin u$.
Let $v = \int_{0}^{y}\frac{dt}{\sqrt{1 - t^2}}$.
Then $y = \sin v$.
Let $c$ be a constant.
$u + v = c$ is a solution of the equation:
$$\frac{dx}{\sqrt{1 - x^2}} + \frac{dy}{\sqrt{1 - y^2}} = 0$$
It suffices to prove that $\sin c = x\sqrt{1 - y^2} + y\... | {
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"timestamp": "2023-03-29T00:00:00",
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Proving inequality $\frac{a}{\sqrt{a+2b}}+\frac{b}{\sqrt{b+2c}}+\frac{c}{\sqrt{c+2a}}\lt \sqrt{\frac{3}{2}}$ In the pdf which you can download here I found the following inequality which I can't solve it.
Exercise 2.1.10 Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that
$$\displaystyle \frac{a}{\sqr... | By C-S $$\left(\sum_{cyc}\frac{a}{\sqrt{a+2b}}\right)^2\leq\sum_{cyc}\frac{a}{(a+2b)(a+2c)}\sum_{cyc}a(a+2c)=\sum_{cyc}\frac{a}{(a+2b)(a+2c)}.$$
Thus, it remains to prove that
$$\sum_{cyc}\frac{a}{(a+2b)(a+2c)}<\frac{3}{2(a+b+c)}$$ or
$$\sum_{cyc}(4a^3b^3+4a^4bc+24a^3b^2c+24a^3c^2b+25a^2b^2c^2)>0.$$
Done!
| {
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"timestamp": "2023-03-29T00:00:00",
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Second order homogenous linear ODE How can I solve this ODE:
$y(x)+Ay'(x)+Bxy'(x)+Cy''(x)+Dx^{2}y''(x)=0$
Can you please also show the derivation.
| I don't think you'll find an "elementary" solution in general. Maple finds a rather complicated solution involving hypergeometric functions:
$$\displaystyle S\, := \,y \left( x \right) ={\it \_C1}\,{\mbox{$_2$F$_1$}(1/2\,{\frac {-d+B+ \sqrt{{d}^{2}+ \left( -2\,B-4 \right) d+{B}^{2}}}{d}},1/2\,{\frac {-d+B- \sqrt{{d}^{... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof of $\frac{d}{dx}e^x = e^x$ I'm working through the proof of $\frac{d}{dx}e^x = e^x$, and trying to understand it, but my mind has gotten stuck at the last step.
Starting with the definition of a derivative, we can formulate it like so:
$$\frac{d}{dx} e^x = \lim_{h \to 0} \frac{e^{x+h}-e^x}{h}$$
After some algebra... | Let's define $b^x$ as
$$ b^x = a_0 + a_1x +a_2 \frac{x^2}{2!} + \ldots =\sum_{k=0}^{\infty} a_k\frac{x^k}{k!}$$
for $x=0$,
$ b^0 =1 $ Thus $a_0=1$
$$ b^{x} = 1 + a_1x +a_2 \frac{x^2}{2!} + \ldots = 1 + \sum_{k=1}^{\infty} a_k\frac{x^k}{k!} \tag 1$$
$$\frac{d}{dx} b^x = \lim_{h \to 0} \frac{b^{x+h}-b^x}{h}=b^x \li... | {
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"source": "stackexchange",
"question_score": "16",
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"answer_id": 8
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How to solve for $ x$ when $x^2 \times a = (x-10)^2 \times b$ I'm trying to figure this one out, and I'm stuck. I'm going:
$x^2 \times a = (x-10)^2 \times b $
$x^2 \times a/b = (x-10)^2 /b $
$a/b = \frac{(x-10)^2} { x^2} / x^2$
$\sqrt{a/b} = (x-10)/x$
I'm not sure how I proceed or if this is even the best ... | $\textbf{Hint}$ : Try to put it into the form $Dx^2 + Ex + F$ and use the quadratic formula to find the solutions. Move cursor over the box for more details.
$ax^2 = b(x - 10)^2$
$ax^2 = b(x^2 - 20x + 100)$
$ax^2 = bx^2 - 20bx + 100b$
$0 = (b - a)x^2 - 20bx + 100b$
Now using the quadratic formula :
$x = \frac{20... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove $\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdot\cdot\cdot+\frac{1}{x^2}<\frac{x-1}{x}$ Please help me for proving this inequality $\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdot\cdot\cdot+\frac{1}{x^2}<\frac{x-1}{x}$
| $$\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdot\cdot\cdot+\frac{1}{x^2}<\frac{x-1}{x}$$
Employ the inequality: $$\frac{1}{a^2}=\frac{1}{a}\cdot\frac{1}{a}<\frac{1}{a}\cdot\frac{1}{a-1}=\frac{1}{a-1}-\frac{1}{a}$$.
In
$$\displaylines{
\frac{1}{{{2^2}}} + \frac{1}{{{3^2}}} + \frac{1}{{{4^2}}} + \cdots + \frac{1}{{{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Sequence Sum {1/2 + 1/4 + 1/6 +...} to infinite I've been told, the following series converges:
$$\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\ldots+\frac{1}{2k}+\ldots$$
I can't get my head around, how to prove this converges; any hints?
| Let
$$S = 1 + \frac{1}{2} + \left( \frac{1}{3} + \frac{1}{4} \right) + \left( \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}\right) + \;...$$
$$[Parentheses \; have \; been \; put \; for \; comparing \; the \; series \; S \; and \; T]$$
Then the given series is simply $2S$. Hence in order to investigate the c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/193515",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Calculate $\lim_{n\to\infty}[(1+x)(1+x^2)(1+x^4)\cdot\cdot\cdot(1+x^{2n})]$, $|x|<1$ Please help me solving $\lim_{n\to\infty}[(1+x)(1+x^2)(1+x^4)\cdot\cdot\cdot(1+x^{2^n})]$, in the region $|x|<1$.
| The product expands out as
$$
(1 + x)(1 + x^2)(1 + x^4)\ldots(1 + x^{2^n}) = \sum_{k=0}^{2^{n+1} -1} x^k = \frac{1 - x^{2^{n+1}}}{1 - x}.
$$
Since $|x| < 1$, this converges to $\frac{1}{1 - x}$ as $n \to \infty$.
| {
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Inequality. $3\left(IA^2+IB^2+IC^2\right) \geq AB^2+BC^2+CA^2$ (Korea 1998) Let $I$ be the incenter of a triangle $ABC$.Prove that:
$$3\left(IA^2+IB^2+IC^2\right) \geq AB^2+BC^2+CA^2.$$
Please help me to improve this kind of inequalities.
Thanks :)
| First note that $IA^2=(p-a)bc/p$ and similarly for $IB, IC$. Substitute to get$$
3abc(\frac{p-a}{ap}+\frac{p-b}{bp}+\frac{p-c}{cp})-a^2-b^2-c^2>=0
$$
Replace $p$ with $(a+b+c)/2$ and get rid of the fractions to arrive at:$$
-a^3 + 2 a^2 b + 2 a b^2 - b^3 + 2 a^2 c - 9 a b c + 2 b^2 c +
2 a c^2 + 2 b c^2 - c^3>=0
$$
S... | {
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Finding the values of the real constants such that the limit exists Find the values of the real constants $c$ and $d$ such that
$$\lim_{x\to 0}\frac{\sqrt{c+dx}-\sqrt{3}}{x}=\sqrt{3}$$
I really have no clue how to even get started.
| $$\lim_{x\to 0}\frac{\sqrt{c+dx}-\sqrt{3}}{x}=\sqrt{3}$$
Since the denominator goes to $0$, the limit cannot exist unless the numerator also goes to $0$. The numerator is $\sqrt{c+dx}-\sqrt{3}$, so that would have to go to $0$ as $x$ goes to $0$. But it goes to $\sqrt{c+d\cdot0} - \sqrt{3}$. Hence $c+d\cdot0$ must b... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Laurent series for $\sin z \sin(1/z)$? How can I find a laurent series for $\sin z \sin(1/z)$ for $z \neq 0$?
Is it possible to multiply two laurent series? I saw that in wikipedia, it's not generally possible.
| Operate formally. Fix finite $z\not=0$. Then each expand each sine into Taylor series:
$$\begin{eqnarray}
\sin(z) \sin\left(\frac{1}{z}\right) &=& \sum_{n=0}^\infty (-1)^n \frac{z^{2n+1}}{(2n+1)!} \sum_{m=0}^\infty (-1)^n \frac{z^{-2m-1}}{(2m+1)!}
\\ &=& \sum_{n=0}^\infty \sum_{m=0}^\infty \frac{(-z^2)^{n-m}}{(2n+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/195636",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What That Mean "In Base 18"? i am a programmer who interest in math , lately in palindromic numbers , so if it's stupid question i am sorry !
i was reading about palindromic numbers in wikipedia , at some point it says
In base 18, some powers of seven are palindromic:
- 7^3 = 111
- 7^4 = 777
- 7^6 = 12... | For example, $12321_{18}=1\cdot 18^4+2\cdot 18^3+3\cdot 18^2+2\cdot 18+1=117649_{10} \\ =1\cdot 10^5+1\cdot 10^4+7\cdot 10^3+6\cdot 10^2+4\cdot 10+9=7^6$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof : $ \sum_{n=1}^{\infty}\frac{1}{2nx^{2n}}=-\frac{1}{2}\ln (1-\frac{1}{x^2})$ How to prove that :
$$ \sum_{n=1}^{\infty}\frac{1}{2nx^{2n}}=-\frac{1}{2}\ln \left(1-\frac{1}{x^2}\right)$$
| We factor out $\frac{1}{2}$ as it is a constant
$$
\sum_{n=1}^{\infty}\frac{1}{2nx^{2n}}=
\frac{1}{2}\sum_{n=1}^{\infty}\frac{1}{nx^{2n}}
$$
Now, the Taylor's series for $-\log (1-x)$ is, for $-1\le x < 1$
$$-\log (1-x)=\sum^{\infty}_{n=1} \frac{x^n}n$$
Thus
$$-\log \left(1-\frac{1}{x}\right)=\sum^{\infty}_{n=1} \frac... | {
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"timestamp": "2023-03-29T00:00:00",
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The exact value of $\sum_{n=1}^{\infty}\frac{n^2+n+1}{3^n}$ What is the value of :
$$\sum_{n=1}^{\infty}\frac{n^2+n+1}{3^n}$$
| In most practical applications, you can just ask Mathematica, and it will tell you it's $\frac{11}{4}$.
If you want to arrive at the formula in a more rigorous way, you can do the following:
Consider the function $f$: $$f(x) = \frac{1}{1 - x} = \sum_{n=0}^\infty x^n$$
An initial observation is that $f(\frac{1}{3}) = \... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Show that the following is indeed a mass function for R.V. $Y$ which can take values $2^n$ and $-2^n$ with probability $\frac{1}{2^{n+2}}$ So I think I have the pieces, just having trouble putting the puzzle together.
$P(Y=2^{n})=P(Y=-2^{n}) = \frac{1}{2^{n+2}}$
\begin{align*}
\sum_{n=0}^{\infty} p_y(y)
&=\sum_{n=0}^{... | Overall you are doing the problem correctly. The probability mass function has been given to you explicitly. Perhaps you were asked to show that it is indeed a pmf, though that is not clear from the question.
If that was asked for, you need to sum the probabilities ("weights") and show that the sum is $1$. The sum of... | {
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Relations between the roots of a cubic polynomial How do I solve the last two of these problems?
The roots of the equation $x^3+4x-1=0$ are $\alpha$, $\beta$, and $\gamma$. Use the substitution $y=\dfrac{1}{1+x}$ to show that the equation $6y^3-7y^2+3y-1=0$ has roots $\dfrac{1}{\alpha+1}$, $\dfrac{1}{\beta+1}$, and $\... | Part 1,
Since , $\frac{1}{\alpha +1}$ is the root of the equation $6y^3-7y^2+3y-1=0\implies $ $$\frac{1}{(\alpha +1)^3}=\frac{7}{(\alpha +1)^2}-\frac{3}{\alpha +1}+1$$
Similar equality follows for $\beta, \gamma$
Part 2,
Product of roots= $\frac{1}{6}\implies \frac{1}{1+\alpha}\frac{1}{1+\beta}\frac{1}{1+\gamma}=\frac{... | {
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Solving $ \sin x + \sqrt 3 \cos x = 1 $ - is my solution correct? I have an equation that I'm trying to solve:
$$ \sin x + \sqrt 3 \cos x = 1 $$
After pondering for a while and trying different things out, this chain of steps is what I ended up with:
$$ \sin x + \sqrt 3 \cos x = 1 $$
$$ \sin x = 1 - \sqrt 3 \cos x $$
$... | You went a little bit astray after $4 \cos^2 x = 2 \sqrt 3 \cos x$, when you divided by $\cos x$: what if $\cos x=0$?
It’s better at that point to bring everything to one side and factor: $4\cos^2x-2\sqrt3\cos x=0$, so $2\cos x(2\cos x-\sqrt3)=0$. Now appeal to the fact that if a product is $0$, at least one of the fac... | {
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"source": "stackexchange",
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Solve $(\sqrt{5+2\sqrt{6}})^{x}+(\sqrt{5-2\sqrt{6}})^{x}=10$.
Solve $(\sqrt{5+2\sqrt{6}})^{x}+(\sqrt{5-2\sqrt{6}})^{x}=10$
I square the both sides and get $(5+2\sqrt{6})^{x}+(5-2\sqrt{6})^{x}=98$. But I don't know how to carry on. Please help. Thank you.
| Let $t_1=(\sqrt{5+2\sqrt6})^x$, and $t_2=(\sqrt{5-2\sqrt6})^x$.
Now the given equation is:
$$\tag 1 t_1+t_2=10$$
But
$$\displaylines{
{t_1}\cdot{t_2}&=& {\left( {\sqrt {5 + 2\sqrt 6 } } \right)^x}{\left( {\sqrt {5 - 2\sqrt 6 } } \right)^x} \cr
&=& {\left[ {\sqrt {\left( {5 + 2\sqrt 6 } \right)\left( {5 - 2\sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/202078",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 3
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Evaluating limit $\lim_{x\to\infty}(x^2-\sqrt{x^4 + 7x^2 + 1})$ The problem is evaluate
$$\lim_{x\to\infty}(x^2-\sqrt{x^4 + 7x^2 + 1})$$
I understand all of the calculus involved, but am having trouble figuring out how to get started with the algebra. I have tried factoring and using conjugates, but the only answer I ... | Let $y = \sqrt{x^4+7x^2+1}$ and $L=\lim (x^2-y)$. Throughout this answer $\lim$ means $\lim_{x\rightarrow \infty}$.
I'll first work on simplifying $y$ as follows. We can take an $x^2$ out of the square root , so that $y= x^2 \sqrt{1+7/x^2 + 1/x^4}$. We can expand the right side out by using the binomial expansion $\sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/204528",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How to integrate $1/(16+x^2)^2$ using trig substitution My answer: $1/128 (\tan^{-1}(x/4) + 4x/(16+x^2)^4)$
| $$
\int\frac{dx}{(16+x^2)^2} = \int\frac{4\sec^2\theta\,d\theta}{(16\sec^2\theta)^2} = \frac{1}{64} \int\frac{d\theta}{\sec^2\theta} = \frac{1}{64}\int \cos^2\theta\,d\theta = \cdots\cdots
$$
\begin{align}
x & = 4\tan\theta \\[6pt]
dx & = 4\sec^2\theta\,d\theta \\[6pt]
16+x^2 & = 16 + 16\tan^2\theta = 16\sec^2\theta
\e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/204961",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Integral calculus What is the integral
$$\int \arccos(z/\sqrt{R^2-x^2})dx$$
This is one of 4 equations for integrating the area of a major sector of a circle within a sphere between limits to find its volume. Two of the functions are easily integrated (first and last), but the above and $\int\arccos(x^2(z/\sqrt{R^2-x^... | To me, it looks like that $\arccos$ part is the source of all ugliness for this problem. I'm not sure using the substitution $x=R\sin\theta$ will simplify this $\arccos$ into something manageable. Therefore, I myself would try to eliminate the $\arccos$ through integration by parts.
$$u=\arccos[z(R^2-x^2)^{-\frac12}]... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/205464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Problems regarding integrals involving Legendre polynomials I am finding difficulty doing this integral involving Legendre polynomials.
$$\int_{-1}^1 x^2 P_{n-1}(x)P_{n+1}(x)dx = \frac{2n(n+1)}{(2n-1)(2n+1)(2n+3)}$$ I have two strategies in my mind both of them have failed to produce results. One is that I could someh... | By Bonnet's formula, we have
\[ (2n-1)xP_{n-1} = nP_n + (n-1)P_{n-2} \]
and
\[ (2n+3)xP_{n+1} = (n+2)P_{n+2} + (n+1)P_n \]
so
\begin{align*}
\int_{-1}^1 x^2 P_{n-1}P_{n+1}\; dx
&= \frac 1{(2n-1)(2n+3)}\int_{-1}^1 \bigl( nP_n + (n-1)P_{n-2}\bigr)\bigl((n+2)P_{n+2} + (n+1)P_n\bigr)\; dx\\
&= \frac 1{(2n-1)(2n+3... | {
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"timestamp": "2023-03-29T00:00:00",
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Find a $4\times 4$ matrix $A$ where $A\neq I$ and $A^2 \neq I$, but $A^3 = I$. Give an example of a $4 \times 4$ matrix where $A \neq I$, $A^2 \neq I$, and $A^3 = I$.
I found a $2 \times 2$ matrix where $A \neq I$ and $A^2 = I$, but this problem is more complex and has me completely stumped.
| Here is a $2\times2$ example
$$
\begin{bmatrix}
-\frac12&\frac{\sqrt{3}}{2}\\
-\frac{\sqrt{3}}{2}&-\frac12
\end{bmatrix}\tag{1}
$$
This works because it is a matrix representation of $e^{2\pi i/3}=-\frac12+i\frac{\sqrt{3}}{2}$; that is, a rotation by $\frac{2\pi}{3}$. Thus, squaring it gives another $2\times2$ example
... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 6,
"answer_id": 2
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Using generating functions find the sum $1^3 + 2^3 + 3^3 +\dotsb+ n^3$ I am quite new to generating functions concept and I am really finding it difficult to know how to approach problems like this. I need to find the sum of $1^3 + 2^3 + 3^3 +\dotsb+ n^3$ using generating functions. How do I proceed about it?
| If you really want to get blown away, consider the following, taken from Aigner's "A Course in Enumeration" (Springer, 2007). Define:
$$
s_m(n) = \sum_{0 \le k < n} k^m
$$
and it's exponential generating function:
$
\begin{align}
\widehat{S}_n(z)
&= \sum_{m \ge 0} s_m(n) \frac{z^m}{m!} \\
&= \sum_{1 \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/209268",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 7,
"answer_id": 4
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how to find the series $x + x^{1 + \frac{1}{2}} + x^{1 + \frac{1}{2}+ \frac{1}{3}} +...$ is convergent. The series $x + x^{1 + \frac{1}{2}} + x^{1 + \frac{1}{2}+ \frac{1}{3}} +...$
is convergent if
(A) $x>e$
(B) $x<e $
(C) $x<1/e$
(D) $x>1/e$
I think the answer is C, but I could not determine the condition... how to... | Nth term of series is given by $u_n=x^{\sum_{i=1}^n \frac{1}{i}}$
Logarithmic test says
Suppose $\sum_{ n \geq 1} a_n $ is a series of positive terms. Suppose
that
$$ \lim_{n \to \infty} n \ln \dfrac{ a_n }{a_{n+1} } = g $$
and if $g>1$ series is convergent and divergent if $g<1$.
now firstly take $x\ge0$, at $x=0$ s... | {
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Monotonicity of $\ln n\over \sqrt{n}$ Is it possible to prove the monotonicity of $(a_n)_n =\frac{\ln n}{\sqrt n}$ without using derivatives?
| For $a_n=\dfrac{\ln{n}}{\sqrt{n}}$ we denote $f_n=\dfrac{1}{a_n^2}=\dfrac{n}{\ln^2{n}}.$ In order to prove that $\left\lbrace a_n \right\rbrace $ is decreasing, we consider
\begin{gather*}
f_{n+1}-f_n=\dfrac{n+1}{\ln^2(n+1)}- \dfrac{n}{\ln^2{n}}=
\dfrac{(n+1)\ln^2{n}-n\ln^2(n+1)}{\ln^2{n} \ \ln^2(n+1)}=\dfrac{A}{B}
\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/210566",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Show that $n^3-n$ is divisible by $6$ using induction As homework, I have to prove that
$\forall n \in \mathbb{N}: n^3-n$ is divisible by 6
I used induction
1) basis: $A(0): 0^3-0 = 6x$ , $x \in \mathbb{N}_0$ // the 6x states that the result is a multiple of 6, right?
2) requirement: $A(n):n^3-n=6x$
3) statement: $A(... | No need for induction.
$$n^3-n=n(n^2-1)=n(n-1)(n+1)$$
which are three consecutive integers. So one must be divisible by 3.
Check for $n=1$: $1^3-1=0=3\cdot 0$
Assume it's true for $n=k$.
If you let $n=k+1$ you get
$$\begin{align*}
(k+1)^3-(k+1)&=k^3+3k^2+2 \\
&=k^3+3k^2+2k\\
&=3\cdot (k^2+k)+(k^3-k)\end{align*}$$
whi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/211121",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 1
} |
When is $1^5 + 2^5 + \ldots + n^5$ a square? When is $1^5 + 2^5 + \ldots + n^5$ a square? I found that this happens sometimes: $n=13$ gives $1001^2$, $n=133$ gives $9712992^2$ and $n=1321$ gives $942162299^2$.
I feel that the identity$$\displaystyle\sum_{i=1}^n i^5 = \tfrac{1}{12}[2n^6+6n^5+5n^4-n^2]$$ will be useful, ... | $$1^5 + 2^5 + ... + n^5 = \frac{1}{12} n^2 (n+1)^2 (2 n^2+2 n-1) $$
Now we need to solve $$\frac{1}{12} n^2 (n+1)^2 (2 n^2+2 n-1) $$ is a perfect square
$$\iff\frac{2 n^2+2 n-1}{12} = k^2$$
[do some thing here] I don't know how to solve more
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/217016",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
"answer_count": 4,
"answer_id": 3
} |
For what values of the variable x does the following inequality hold: $\ \frac{4x^2}{\Bigl(1-\sqrt{\ 1\ +2x}\Bigr)^2} <
2x+9$
... IMO-1960
| Maybe you wanna write everything as
$$\ \frac{((2x+1)-2)(2x+1)+1}{\Bigl(1-\sqrt{2x+1}\Bigr)^2} <
2x+1+8$$
Then, let's denote $2x+1=y$ that yields
$$\ \frac{(y-2)y+1}{\Bigl(1-\sqrt{y}\Bigr)^2} <
y+8$$
$$\ \frac{(1-y)^2}{\Bigl(1-\sqrt{y}\Bigr)^2} <
y+8$$
$$\ \left(\frac{(1-\sqrt{y})(1+\sqrt{y})}{1-\sqrt{y}}\right)^2 <
y+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/219127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
What are the steps to simplify the following modulus expression? I have no clue how to do this exactly. Is there a systematic way of doing this or you just have to do it by trial and error?
$n^2 \equiv 9 \pmod {72}$
to $n \equiv a \pmod b$?
| Note that $72=8\cdot 9$ and that $n^2\equiv 9\pmod{72}$ implies (and is in fact equivalent to) $n^2\equiv 9\pmod 9$ and $n^2\equiv 9\pmod 8$.
The first simplifies to $n^2\equiv 0\pmod 9$, i.e. $3^2|n^2$. This is equivalent to $3|n$. The other equation simplifieds to $n^2 \equiv 1\pmod 8$. You may try these few cass by ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/222918",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 4
} |
Proving that $|CA|+|CB|=2|AB|$ in a general $ABC$ triangle How in this situation (presented in image) can I prove that $|CA|+|CB|=2|AB|$?
| Here's a calculatory approach:
Call $AE=x$ and $BD=y$ and let the angles at $A$, $B$, $C$ be $\alpha$, $\beta$, $\gamma$, respectively.
Now, the area of the triangle $ABC$ equals $$\frac 12 (a+x) (b+y) \sin \gamma.$$
But it is also the sum of the three smaller regions:
$$ab\sin\gamma +\frac 12 ay\sin\beta +\frac 12 xb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/227129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
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Fractions in Questions and Answers
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