Short prove by induction

cyb3r · #1 $Old$ November 16th, 2008, 11:21 AM

I need to prove the following for one of my solutions in order to solve a problem:

For $n \in\mathbb{N}$ if $n>=3^3$ then $3^n > 79n^2$

Thank you in advance.

clic-clac · #2 $Old$ November 16th, 2008, 11:57 AM

Have you checked that $3^{3^{3}}>79.(3^{3})^{2}$ ?

Then, let $n \geq 3^{3}$ be an integer, and assume that $3^{n}>79n^{2}$ (induction hypothesis)

$79(n+1)^{2}=79n^{2}+79(2n)+79$

Is it true that, if $n\geq 3^{3}\$ , then $\ 2n\leq n^{2}$ and $1\leq n^{2}$ ?

If it's the case, $79n^{2}+79(2n)+79 \leq 79n^{2}+79n^{2}+79n^{2}=3(79n^{2})$

What can we conclude using the induction hypothesis?