Help with a proof by induction

• Oct 29th 2012, 06:01 PM
Walshy
Help with a proof by induction
Prove the following using the principle of mathematic induction.
n! > n^2, for all integers n >= 4.

I am especially having trouble with the inductive step. Thanks for your help.
• Oct 29th 2012, 06:06 PM
Prove It
Re: Help with a proof by induction
Quote:

Originally Posted by Walshy
Prove the following using the principle of mathematic induction.
n! > n^2, for all integers n >= 4.

I am especially having trouble with the inductive step. Thanks for your help.

Can you please show us what you have done? Then we can set you in the right direction.
• Oct 29th 2012, 06:09 PM
Walshy
Re: Help with a proof by induction
Basis Step: 4!>4^2, 24>16 - that works.
Inductive step: Assume k! >k^2
Goal: (k+1)! > (k+1)^2
= k+1(k!) > k^2+2k+1
stuck here.
• Oct 29th 2012, 06:15 PM
Prove It
Re: Help with a proof by induction
Quote:

Originally Posted by Walshy
Basis Step: 4!>4^2, 24>16 - that works.
Inductive step: Assume k! >k^2
Goal: (k+1)! > (k+1)^2
= k+1(k!) > k^2+2k+1
stuck here.

The way you set out these problems needs work - you need to start on one side of your statement and go through a series of arguments to get to the other side. But anyway, your logic is right at least...

\displaystyle \begin{align*} ( k + 1 ) ! &= (k + 1)k! \\ &> (k + 1)k^2 \\ &> (k + 1)^2 \textrm{ since } k^2 > k + 1 \end{align*}

It's up to you to prove \displaystyle \begin{align*} k^2 > k + 1 \end{align*} in our region :)