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Triangles Within Pentagons

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Diagrams of pentagons growing
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 $T_{3n-1}$