How to start the induction step for an inequality.

Question

If $ n $ is a natural number and $n \geq 4$ then $3^n > 2n^2 +3n$. Note* The inequality is false when $ n=1,2, $ and $ 3 $. I understand how to prove the base case. I'm having trouble proving the induction step for $ P(k+1) $ . I'm not sure how to get $ (k+1) $ on both sides to prove the inequality is true. Any ideas on how to get started? Thanks.

Answer 1

Suppose that $3^n > 2n^2 + 3n$ for some natural $n \ge 4$. Then you want to prove that $3^{n+1} > 2(n+1)^2 + 3(n+1)=2n^2+4n+2+3n+3=2n^2+3n+4n+5$.

Hence, you just need to prove that $2(3)^n>4n+5 $ for all $n \ge 4$, after you have completed your base case.