I need to prove the following for one of my solutions in order to solve a problem:
For $\displaystyle n \in\mathbb{N}$ if $\displaystyle n>=3^3$ then $\displaystyle 3^n > 79n^2$
Thank you in advance.
Have you checked that $\displaystyle 3^{3^{3}}>79.(3^{3})^{2}$?
Then, let $\displaystyle n \geq 3^{3}$ be an integer, and assume that $\displaystyle 3^{n}>79n^{2}$ (induction hypothesis)
$\displaystyle 79(n+1)^{2}=79n^{2}+79(2n)+79$
Is it true that, if $\displaystyle n\geq 3^{3}\ $, then $\displaystyle \ 2n\leq n^{2}$ and $\displaystyle 1\leq n^{2}$ ?
If it's the case, $\displaystyle 79n^{2}+79(2n)+79 \leq 79n^{2}+79n^{2}+79n^{2}=3(79n^{2})$
What can we conclude using the induction hypothesis?