I'm sharpening my math skills by myself since school never motivated me to do it, so right now I'm doing some induction exercises.
I just solved one and I'm not sure if it's right since it doesn't fit the solution I was given.
I skipped the base case.
$ 3^n > n^2 , n > 2 \\ 3^k > k^2 \\ 3^{k+1} > (k+1)^2 \\ 3^k > k^2 | \cdot 3 \\ 3^{k+1} > 3k^2 \\ 3k^2 > k^2 + 2k + 1 \\ 2k^2 > 2k+1 \\ 2k^2 - 2k > 1 \\ 2k(k+1) > 1 \\ \begin{cases}k > 2 => k+1 > 1 \\2k > 1\end{cases} \implies 3^{k+1} > (k+1)^2 $
The solution I saw was more bottoms-up and mine seems to be solved top-down anyway that's why I'm wondering.
Sorry for my Tex just learned some to hack this together. Thanks for taking time to read and answer my newbie doubt.
We can streamline the development as follows. First, we establish a base case. For $n=1$, this is $3^1>1^2$. Now, we assume that for some number $n\ge 2$ we have
$$3^n>n^2$$
Then, proceeding by induction and noting that for $n\ge 2$, $n^2\ge 2n>1$, we have
$$\begin{align} 3^{n+1}&=(3)\,3^n\\\\ &>3n^2\\\\ &=n^2+n^2+n^2\\\\ &>n^2+2n+1\\\\ &=(n+1)^2 \end{align}$$
And we are done!