The diagram above shows the pentagon growing in a systematic way to
produce the pentagonal numbers. The equations below
describe this growth:
P2
=
P1 + 4
=
1+4
=
1 + 3 ×1 + 1
P3
=
P2 + 7
=
1+4+7
=
1 + (3 ×1 + 1) + (3 ×2 + 1)
P4
=
P3 + 10
=
1 + (3 ×1 + 1) + (3 ×2 + 1) + (3 ×3 + 1)
Can you find a general rule for Pn?
By writing a formula for the nth triangular number
Tn, show that all pentagonal numbers are one
third of a triangular number and prove that the triangular
numbers involved are all of the form T3n-1
