For $n \in Z^{\geq 0}$, define $g_n = 2^{2^n} + 1$. Show that $g_0\cdot g_1\cdots g_{n-1} = g_n -2$.
105671
Let $g_n= {2^2}^n +1 $. Prove $g_0 · g_1 · · · g_{n−1} = g_{n} − 2$
62254
Show $F_{n+1} \cdot F_{n-1} = F_n^2 + (-1)^n$ for all $n \in \mathbb{N}$
51730
How to prove this inequality $3^{n}\geq n^{2}$ for $n\geq 1$ with mathematical induction?
21217
$G$, group. $a,b\in G$. Show that $|a|=|a^{-1}|; |ab|=|ba|$, and $|a|=|cac^{-1}|, \forall c\in G$.
120642
Let $g : \Bbb N \times \Bbb N \to\Bbb N \times \Bbb N$ defined as $g(m,n) = (m + n,m - n)$
6453
Let $n \geq 1$ be an odd integer. Show that $D_{2n}\cong \mathbb{Z}_2 \times D_n$.
101057
Show that $1^{p-1} + 2^{p-1} +\ldots + (p-1)^{p-1} \equiv -1 \mod p$
84213
How to show $n\sum_{i=1}^n {x_i^2} \ge (\sum_{i=1}^n{x_i})^2$
12781
Prove the following inequality $\sqrt[n]{x_1x_2...x_n}\ge (n-1)$
70128
Prove that $(fg)^{(n)} = \sum_{k=0}^n \binom{n}{k}f^{(k)}(x)g^{(n-k)}(x)$
51769
If $x,y,z\in[-1,1]$ and $1+2xyz\geq x^2+y^2+z^2$, then can we infer $1+2(xyz)^n\geq x^{2n}+y^{2n}+z^{2n}$?
5708
Prove that if $k\ge 1\,$ and $\,a^k\equiv 1\pmod{\! n}$ then $\gcd(a,n)=1$
5927
How to prove $a_1^m + a_2^m + \cdots + a_n^m \geq \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}$
33300
Showing order of $Z (G) =1$ or $pq$
104855
Prove that $N \subset Z(G)$
11003
Proving that $G/N$ is an abelian group
107071
Prove $\sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n - 1}$ using the binomial theorem