Pentagonal numbers are truncated triangular numbers

Pentagonal numbers are truncated triangular numbers. You can take the diagram that illustrates the nth pentagonal number and warp it into the base of the image that illustrates the (2n − 1)st triangular number. If you added a diagram for the (n − 1)st triangular number to the bottom of the image on the right, you’d have a diagram for the (2n − 1)st triangular number.

In short,

P_n = T_{2n − 1} − T_n.

This is trivial to prove algebraically, though the visual proof above is more interesting.

The proof follows immediately from the definition of pentagonal numbers

P_n = (3n² − n)/2

and triangular numbers

T_n = (n² − n)/2.

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