query: Help me prove that $\frac{(2n)!}{n!^2}$ is a natural number by induction.
positive_passages.docid: 20691
positive_passages.text: prove that $\frac{(2n)!}{(n!)^2}$ is even if $n$ is a positive integer
negative_passages.docid: 26901
negative_passages.text: induction proof: $\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$
negative_passages.docid: 79145
negative_passages.text: Proving by induction: $2^n > n^3 $ for any natural number $n > 9$
negative_passages.docid: 40348
negative_passages.text: Proving by mathematical induction: $1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{n}}>2(\sqrt{n+1}-1)$
negative_passages.docid: 90789
negative_passages.text: Use induction to prove that that $8^{n} | (4n)!$ for all positive integers $n$
negative_passages.docid: 145519
negative_passages.text: Prove that for any natural number $n>1$, we have $(2n)!<(n(n+1))^n$
negative_passages.docid: 78583
negative_passages.text: Help with proof using induction: $1 + \frac{1}{4} + \frac{1}{9}+\cdots+\frac{1}{n^2}\leq 2-\frac{1}{n}$
negative_passages.docid: 51928
negative_passages.text: Use an induction argument to prove that for any natural number $n$, the interval $(n,n+1)$ does not contain any natural number.
negative_passages.docid: 81842
negative_passages.text: How to prove by induction that $\frac{a^n+b^n}{2}\geq\left(\frac{a+b}{2}\right)^n$?
negative_passages.docid: 88055
negative_passages.text: Use induction to prove the following: $1! + 2! + .... + n! \le (n + 1)!$
negative_passages.docid: 62177
negative_passages.text: Prove that: $2^n < n!$ Using Induction
negative_passages.docid: 22073
negative_passages.text: Prove by induction that $n^2<n!$
negative_passages.docid: 54059
negative_passages.text: Showing that $\frac{z^n}{1+z^{2n}}$ is real for $|z|=1$.
negative_passages.docid: 16798
negative_passages.text: Prove the inequality $n! \geq 2^n$ by induction
negative_passages.docid: 78726
negative_passages.text: Prove via mathematical induction that $4n < 2^n$ for all $ n≥5$.
negative_passages.docid: 114212
negative_passages.text: How to use Mathematical Induction to prove $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \cdots + \frac{1}{n(n + 1)} = \frac{n}{n + 1}$?
negative_passages.docid: 27304
negative_passages.text: Proving $ 1+\frac{1}{4}+\frac{1}{9}+\cdots+\frac{1}{n^2}\leq 2-\frac{1}{n}$ for all $n\geq 2$ by induction