You may also like

problem icon

Plum Tree

Label this plum tree graph to make it totally magic!

problem icon

Magic W

Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.

problem icon

2-digit Square

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

Triangles Within Pentagons

Age 14 to 16 Challenge Level:

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 $T_{3n-1}$