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,$$a_k=(k-1)(k+1)+2k+1=\cdots$$