Proof by induction that $1^2 + 2^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6}$
26901
induction proof: $\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$
9435
Prove that $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$.
122019
Prove by mathematical induction that $2^{3^n}+1$ is divisible by $3^{n+1}$
136994
Proving :$\frac{1}{2ab^2+1}+\frac{1}{2bc^2+1}+\frac{1}{2ca^2+1}\ge1$
87941
Prove by mathematical induction that $\forall n \in \mathbb{N} : \sum_{k=1}^{2n} \frac{(-1)^{k+1}}{k} = \sum_{k=n+1}^{2n} \frac{1}{k} $
70625
Prove By Induction $1 + 1/2 + 1/3 + ... + 1/(2^n) ≥ 1 + n/2$
85941
Proof by induction that $n^3 - n$ is divisible by $6$
57491
Double induction example: $ 1 + q + q^2 + q^3 + \cdots + q^{n-1} + q^n = \frac {q^{n+1}-1}{q-1} $
131565
Proof by induction of summation inequality: $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots+\frac1{2^n}\ge 1+\frac{n}2$
156356
Proof that $x^n+\frac{1}{x^n}\in \mathbb Z$ by complete induction
115912
Summation inductional proof: $\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\ldots+\frac{1}{n^2}<2$
121544
Proof by Induction: $2(\sqrt{n+1} - \sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n}-\sqrt{n-1})$
65315
Showing $1+2+\cdots+n=\frac{n(n+1)}{2}$ by induction (stuck on inductive step)
17491
Proof by Induction: Solving $1+3+5+\cdots+(2n-1)$
48503
Proving : $ \bigl(1+\frac{1}{n+1}\bigr)^{n+1} \gt (1+\frac{1}{n})^{n} $
146200
proof by induction: sum of binomial coefficients $\sum_{k=0}^n (^n_k) = 2^n$
15164
Proof of $\sum^{2N}_{n=1} \frac{(-1)^{n-1}}{n} = \sum^{N}_{n=1} \frac{1}{N+n}$