Question

Prove by mathematical induction

2n>(n+1)! For all integars n>=2

Answer 1

The correct inequality is $2^n<(n+1)!$

Basis of induction: n=2.

$2^n = 4 < (2+1)! = 6$ is true.

Assume now that the inequality is proved for n=N-1. Then it is correct for n=N:

$(N+1)! = (N+1)\cdot N!>(N+1)\cdot 2^{N-1} = 2\cdot 2^{N-1}=2^N$

Therefore, the statement is proved by the principle of the mathematical induction.