Prove that ${\sqrt {n} }^{\sqrt {n+1}} > {\sqrt {n+1}}^{\sqrt {n}};n=7,8,9...$
100375
Prove that ${\sqrt {n} }^{\sqrt {n+1}} > {\sqrt {n+1}}^{\sqrt {n}}$.
66387
Prove $(x_1+.....+x_n)^2 \leq n(x_1^2 + .....+x_n^2)$
145891
Prove that $n! > \sqrt{n^n}, n \geq 3$
66016
Prove $3^{2n+1} + 2^{n+2}$ is divisible by $7$ for all $n\ge0$
120093
Proof of inequality $2(\sqrt{n+1}-\sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n} - \sqrt{n-1})$ using induction
78985
Prove that: $\sqrt{2\sqrt{3\sqrt{4\cdots\sqrt{n}}}}<3,\,\forall n\in\mathbb N.$
26912
Prove by induction that for all $n \geq 3$: $n^{n+1} > (n+1)^n$
34491
Is there a way to show that $\sqrt{p_{n}} < n$?
125040
Show by induction : $n^7-n$ is a multiple of 7
99428
Prove that $3^{2n-1} + 2^{n+1}$ is always a multiple of $7$.
89175
Proof of $(a+b)^{n+1}$
97656
Finding $a_n$ if $a_0=a_1=1,a_{n+1}=n(a_n+a_{n-1})\ \ (n\ge 1).$
57095
How do I prove the sequence $x_{n+1}=\sqrt{\alpha +x_n}$, with $x_0=\sqrt \alpha$ converges? ( boundedness?)
85977
Show that if $\sqrt{a}+\sqrt{b}\neq 0$, then $\mathbb{Q}(\sqrt{a}+\sqrt{b}) = \mathbb{Q}(\sqrt{a},\sqrt{b}), \forall a,b\in\mathbb{Q}$
5797
Limit of the sequence $a_n=\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}$
105110
Why does $\sqrt{n\sqrt{n\sqrt{n \ldots}}} = n$?
12729
Proving $9^n - 8n - 1$ is divisible by $8^2$ for $n\ge 0$?