# How to prove a recursive equation with mathematical induction?

Let the sequence $$\{a_n\}_{n\ge0}$$ be recursively defined as follows: $$a_0=0,\,\forall k\ge1(a_k=a_{k-1}+2k+1).$$ Using mathematical induction, prove that this sequence satisfies the following equation:$$\forall n\ge0(a_n=n(n+2)).$$

I have done the basis step as shown below, but still struggling to solve the inductive step.

Basis step: for $$n=0$$, $$a_0=0=0(0+2)$$.

We want to prove if $$a_{k-1}=(k-1)(k+1)$$ then $$a_k=k(k+2)$$. By the inductive hypothesis,$$a_k=(k-1)(k+1)+2k+1=\cdots$$