For every natural number $n$, prove that $n^n > 1 \cdot 3 \cdot 5 \cdot 7 \cdots (2n-1)$.
133017
Prove that $n^n>1\times3\times5\times7\times \dots\times(2n-1)$
88995
Prove that $a^n-b^n = (a-b)(a^{n-1} + a^{n-2}b + \cdots + b^{n-1})$.
83752
If $p + q = 1$ prove that for any natural $n, m$ following is true: $(1 - p^n)^m + (1 - q^m)^n \ge 1$
66016
Prove $3^{2n+1} + 2^{n+2}$ is divisible by $7$ for all $n\ge0$
152577
Proving $4^n > n^4$ holds for $n\geq 5$ via induction.
95061
Proving that $2^{2\cdot 3^{n-1}}\equiv 1+3^n\pmod{3^{n+1}}$ for every natural $n$
141909
Prove ${2}^{n}\cdot(2m+1)-1 $ is invertible
116356
Combinatorial proof of $2 \cdot 3^0 + 2 \cdot 3^1 + 2 \cdot 3^2 + \cdots 2 \cdot 3^{n-1} = 3^n-1$
81136
Prove by induction that... $1+3+5+7+...+(2n+1)=(n+1)^2$ for every $n \in \mathbb N$
63416
Prove $(n^5-n)$ is divisible by 5 by induction.
125040
Show by induction : $n^7-n$ is a multiple of 7
82774
Prove by mathematical induction that $n^3 - n$ is divisible by $3$ for all natural number $n$
21612
How to prove this form of $n$?
78726
Prove via mathematical induction that $4n < 2^n$ for all $ n≥5$.
52032
Prove that ${n^5 - n}$ is divisible by 5
27449
Product of Divisors of some $n$ proof
95889
Prove that $\gcd(n! + 1, 3n! + 1) = 1$, $\forall n \in \mathbb{N}$, $n \geq 2$.