# Inductive Proof

• August 2nd 2011, 02:57 PM
veronicak5678
Inductive Proof
For which integers is 3^n > n^3?

it seems to be for all but 3, but I am stuck on the proof.
Assume 3^n > n^3
n>3
Then
3^(n+1) > 3*n^3
Now what?
• August 2nd 2011, 04:14 PM
Plato
Re: Inductive Proof
Quote:

Originally Posted by veronicak5678
For which integers is 3^n > n^3?
it seems to be for all but 3, but I am stuck on the proof.
Assume 3^n > n^3 if n>3

$\text{If }n\ge 4\text{ then }3n^2+3n+1\le 48+12+1=61<3^n~.$

So that means that $(n+1)^3\le n^3+3^n~.$

You must find out how that helps you. Hint $2<3$.
• August 2nd 2011, 04:49 PM
Re: Inductive Proof
Quote:

Originally Posted by veronicak5678
For which integers is 3^n > n^3?

it seems to be for all but 3, but I am stuck on the proof.
Assume 3^n > n^3
n>3
Then
3^(n+1) > 3*n^3
Now what?

Another way to consider...

You want to show

$3^{n+1}>(n+1)^3$

Note that $3^1>1^3,\;\;\;3^2>2^3,\;\;\;3^3=3^3$

Hence n>3 for the proof...

If $3^n>n^3$, then $3^{n+1}>3n^3$

If we can show that $3n^3>(n+1)^3$, for n>3, the proof is complete.

Since we know n>3, then

$3n^3>n^3+3n^2+3n+1\;\;?$

$2n^3>3n^2+3n+1\;\;?$

Now use the fact that if n>3, then $2n^3>2(3)n^2$ and continue
• August 2nd 2011, 05:40 PM
veronicak5678
Re: Inductive Proof

What do you mean by the question marks?
We know n > 3, so 3n^3 > 3*81
• August 3rd 2011, 02:01 AM
Re: Inductive Proof
Quote:

Originally Posted by veronicak5678

What do you mean by the question marks?
We know n > 3, so 3n^3 > 3*81

The purpose of the question marks is...

We are asking if $3n^3>n^3+3n^2+3n+1,\;\;if\;\; n>3$

Because if that's true, then $3^{n+1}>(n+1)^3$

Proof By Induction requires that you show $3^{n+1}>(n+1)^3\;\;\;if\;\;\;3^n>n^3$

$2n^3\;\;is\;greater\;than\;\;\;3n^2+3n+1,\;\;n>3$

Now if n>3, then $2n^3>2(3)n^2$

and you can finish by showing than $6n^2>3n^2+3n+1$
using the same reasoning,
since $2n^3>6n^2,\;\;\;n>3$
• August 3rd 2011, 03:02 AM
Swlabr
Re: Inductive Proof
Quote:

Originally Posted by veronicak5678
For which integers is 3^n > n^3?

it seems to be for all but 3, but I am stuck on the proof.
Assume 3^n > n^3
n>3
Then
3^(n+1) > 3*n^3
Now what?

Just to point out - you also need to consider the negative numbers, which won't be covered by your induction. That said, that aren't too hard if you thing about it correctly...(Where does $3^{-n}$ lie? What about $(-n)^3$?)
• August 3rd 2011, 10:47 AM
veronicak5678
Re: Inductive Proof
Got it. Thanks very much, everyone!