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Questions tagged [faq]

This is meant for questions which are generalized forms of questions which get asked frequently. See tag details for more information.

When you add a question marked faq, please also update the list of questions: List of Generalizations of Common Questions

The question which prompted this: Coping with *abstract* duplicate questions.

Note: Even though one might argue that tagging a question as faq should be enough, and there is no need to update the above list, updating the above list will serve to bring this policy back to attention and help raise awareness periodically.

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Show that if $n>2$, then $(n!)^2>n^n$.

Show that if $n>2$, then $(n!)^2>n^n$. My work: I tried to apply induction. So, at the induction step, I need to prove, $n^n>(n+1)^{n-1}$ Here, I tried to use induction again without any luck. I also took log of both sides, but I did not get…
Hawk
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How to solve homogeneous linear recurrence relations with constant coefficients?

Consider a sequence $(a_n)_{n\in\mathbb N}$ defined by $k$ initial values $(a_1,\dots,a_k)$ and $$a_{n+k}=c_{k-1}a_{n+k-1}+\dots+c_0a_n$$ for all $n\in\mathbb N$. What are some ways to get closed forms for $a_n$? What are some ways of rewriting…
Simply Beautiful Art
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calculating $a^b \!\mod c$

What is the fastest way (general method) to calculate the quantity $a^b \!\mod c$? For example $a=2205$, $b=23$, $c=4891$.
minasteris
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For a finite group of order $2n$ does there exist $x$ such that $x\ast x=e$?

Let $ (G,\ast)$ be a group with identity $e$ and cardinality $2n$ for some $n\in\omega$. Then, does there exist $x\in G$ such that $x\ast x=e$ and $x\neq e$?
John. p
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New Two Children problem

The well known "Two Children problem" is answered in " In a family with two children, what are the chances, if one of the children is a girl, that both children are girls?" What about this variant: M. Smith says: I have two children and at least one…
Jean-Pierre
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What is the error in this fake proof which uses series to show that $1=0$?

A common "trick" for obtaining a closed form of a geometric series is to define $$ R := \sum_{k=0}^{\infty} r^k, $$ then manipulate the series as follows: \begin{align} R - rR &= \sum_{k=0}^{\infty} r^{k} + \sum_{k=0}^{\infty} r^{k+1} \\ &= (1 + r +…
Xander Henderson
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Rigorous proof that $\int_{\Omega}X\;dP=\int_{-\infty}^{\infty}xf(x)\;dx$

I'm trying to prove rigorously that $\int_{\Omega}X\;dP=\int_{-\infty}^{\infty}xf(x)\;dx$. Where $f$ is the pdf of the random variable $X$. I can't find a proof on the wikipedia article, or if it's there then it's disguised enough that I can't…
Thoth
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What does matrix multiplication have to do with scalar multiplication?

Why are matrix and scalar multiplication denoted the same way and treated as the same operation in standard mathematical notation? This is always a source of confusion for me because they have completely different properties (specifically…
dsimcha
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How Do I Compute the Eigenvalues of a Small Matrix?

If I have a $2\times 2$ or $3\times 3$ matrix, how should I go about computing the eigenvalues and eigenvectors of the matrix? NB: I am making this question to provide a unified answer to questions about eigenvalues of small matrices so that all of…
Stella Biderman
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Probability for roots of quadratic equation to be real, with coefficients being dice rolls.

I really need help with this question. The coefficients $a,b,c$ of the quadratic equation $ax^2+bx+c=0$ are determined by throwing $3$ dice and reading off the value shown on the uppermost face of each die, so that the first die gives $a$, the…
CJS
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Converting multiplying fractions to sum of fractions

I have the next fraction: $$\frac{1}{x^3-1}.$$ I want to convert it to sum of fractions (meaning $1/(a+b)$). So I changed it to: $$\frac{1}{(x-1)(x^2+x+1)}.$$ but now I dont know the next step. Any idea? Thanks.
Adam Sh
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When does $p^2$ divide $an^k + bp$?

In the ongoing effort of dealing with abstract duplicates. This question is about the lemma: Lemma Let $k \ge 2$, $p$ prime and $a$ coprime to $p$. Then $$p^2\!\mid a n^k+ bp\iff p\mid n,b.$$ This lemma answers the following kinds of…
Trevor Gunn
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Differentiation of $x^{\sqrt{x}}$, how?

The answer is (I think) $x^{\sqrt{x}-0.5} (1+0.5\ln(x))$, but how?
aiao
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Find $6^{1000} \mod 23$

Find $6^{1000} \mod 23 $ Having just studied Fermat's theorem I've applied $6^{22}\equiv 1 \mod 23 $, but now I am quite clueless on the best way to proceed. This is what I've tried: Raising everything to the $4th$ power I have $$6^{88} \equiv 1…
Mr. Y
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Proof of convergence of a recursive sequence

How do I prove that $x_{n+2}=\frac{1}{2} \cdot (x_n + x_{n+1})$ $x_1=1$ $x_2=2$ is convergent?
geek4079
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