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## Homework Statement

Prove that 3^n>n^4 for all n in N , n>=8

## Homework Equations

## The Attempt at a Solution

Base case: 3^8>8^4

Inductive step

Assume 3^n>n^4. Show 3^n+1>(n+1)^4

I tried a lot of approaches to get from the inductive hypothesis to what I want to show

Ex:

3^n>n^4

3^n+1>3n^4

3n^4>(3n^4)-3=3(n^4-1)=3((n^2)-1)((n^2)+1)=3(n+1)(n-1)(n^2+1)>3(n+1)(n-1)(n^2-1)

=3(n+1)(n-1)(n+1)(n-1)=3((n+1)^4-(n-1)^4)

It looks like I went too far

My other approach is this. It looks a little bit crazy, but I think it works

3^n+1>n^4+2*3^n

I will show that n^4+2*3^n grows faster than (n+1)^4 by using limits and loopitals rule.

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