# Proof by induction with inequalities

Prove $$5n + 6 \leqslant n^2$$ holds for all $$n \geqslant N$$ by induction. Here $$N$$ is the answer you get in (a).

For (a) I got $$6$$ and I proceeded as follows:

Base case: $$n = 6$$: $$5(6)+6 \leqslant 6^22$$, $$36 \leqslant 36$$ therefore base case is true.

Assume $$5k + 6 \leqslant k^2$$ for $$k \geqslant 6$$.

Induction: $$5(k+1) + 6 \leqslant (k+1)^2$$ is true $$5k + 5 + 6 \leqslant k^2 + 2k +1$$ I'm some how confused as to what I need to do next.

You are almost done. Now note that, since $$5k+6\leqslant k^2$$, then $$5k+5+6=5k+6+5\leqslant k^2+5$$ and that $$5\leqslant2k+1$$, since $$k\geqslant6$$. So,$$5k+5+6\leqslant k^2+2k+1=(k+1)^2.$$