Challenge Level

The diagram above shows the pentagon growing in a systematic
way to produce the pentagonal numbers. The equations below describe
this growth:

$$\begin{eqnarray} P_2 &=&
P_1 + 4 \\ &=&1+4 \\ &=&1 + 3 \times 1 + 1\\
P_3&=& P_2 + 7 \\ &=& 1+4+7 \\ &=& 1 + (3
\times 1 + 1) + (3 \times 2 + 1) \\ P_4 &=& P_3 + 10 \\
&=& 1 + (3 \times 1 + 1) + (3 \times 2 + 1) + (3 \times 3 +
1) \\ \end{eqnarray}$$

Can you find a general rule for $P_n$?

By writing a formula for the nth triangular number $T_n$, show
that all pentagonal numbers are one third of a triangular number
and prove that the triangular numbers involved are all of the form
$T_{3n-1}$