2010 ACJC P2 Q3

The sequence of real numbers \({{b}_{1}}\), \({{b}_{2}}\), \({{b}_{3}}\), …. is such that \({{b}_{1}}=4\) and \({{b}_{n}}={{b}_{n-1}}+2n\) for all \(n\in {{\mathbb{Z}}^{+}}\), \(n\ge 2\).

Using the method of difference, show that \({{b}_{n}}={{n}^{2}}+n+2\) for all \(n\in {{\mathbb{Z}}^{+}}\).[3]