Thread: Help with a proof by induction

1. 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.

2. Re: Help with a proof by induction

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.

3. 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.

4. Re: Help with a proof by induction

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