The pentagonal numbers p1, p2, p3, pk, …, are the integers that count the number of dots i...

The pentagonal numbers p1, p2, p3, pk, …, are the integers that count the number of dots in k nested pentagons, as shown in the following figure.

Show that p1= 1 and pk = pk− 1 + (3k − 2) for k ≥ 2. Conclude that and evaluate this sum to find a simple formula for pn.