If n is a positive integer, then prove that $2^n \gt 1 + n\;\sqrt{{2}^{n-1}}$ ; $n\gt 1$
5043
Show that $2^n>1+n\sqrt{2^{n-1}}$
59475
Show that $\sqrt{n+1}-\sqrt{n}\to0$
121544
Proof by Induction: $2(\sqrt{n+1} - \sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n}-\sqrt{n-1})$
19046
Show that if $a$ and $b$ are positive integers with $(a,b)=1$ then $(a^n, b^n) = 1$ for all positive integers n
24469
Prove $ 2<(1+\frac{1}{n})^{n}$
12781
Prove the following inequality $\sqrt[n]{x_1x_2...x_n}\ge (n-1)$
10629
How to prove $4(n!)>2^{n+2}$ for $ n\geq 4$ with induction
120093
Proof of inequality $2(\sqrt{n+1}-\sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n} - \sqrt{n-1})$ using induction
70625
Prove By Induction $1 + 1/2 + 1/3 + ... + 1/(2^n) ≥ 1 + n/2$
24403
Prove that $ n < 2^{n}$ for all natural numbers $n$.
75373
Show that $2^n>n$ for $n\in \mathbb{N}$
118244
An $n$th root inequality: $\sqrt[n]{n} < 1 + \sqrt{2/n}$
32149
Proof of the inequality $(x+y)^n\leq 2^{n-1}(x^n+y^n)$
96097
Let $k$ be an odd positive integer. Prove that $(1+2+3...+n)|(1^k+2^k+3^k+...+n^k)$ for all positive integers $n$.
148455
Show that if $(a,n)=d>1$ and $k$ is any positive integer, then $a^k\not\equiv 1\pmod{n}.$
26912
Prove by induction that for all $n \geq 3$: $n^{n+1} > (n+1)^n$
27580
Use mathematical induction to prove $ (2n)!\geq 2^n(n!)^2$ for $n \in \mathbb{N}$