prove-by-mathematical-induction-n-n-n-

Question Number 9790 by tawakalitu last updated on 04/Jan/17

prove by mathematical induction

n! ⩽ n^{n}

Commented by FilupSmith last updated on 04/Jan/17

n = 1

1! ⩽ 1^{1}

n = 2

2! ⩽ 2^{2}

2 ⩽ 4

(n + 1)! ⩽ {(n + 1)}^{n + 1}

(n + 1) \cdot n! ⩽ {(n + 1)}^{n + 1}

n! ⩽ {(n + 1)}^{n}

\underset{n t i m e s}{1 \times 2 \times \dots \times n} ⩽ \underset{n t i m e s}{(n + 1) \times \dots \times (n + 1)}

\forall k \in (1, n), (n + 1) > k

Commented by tawakalitu last updated on 05/Jan/17

Thank you sir . God bless you .

Commented by tawakalitu last updated on 05/Jan/17

Thank you sir . God bless you .