Triangular, Pentagonal, etc... Numbers

Polygonal numbers represent the number of dots that can be arranged in the shape of a regular polygon. The formula for the nth M-gonal number is $M_n = \frac{n[(M-2)n - (M-4)]}{2}$.

Key formulas to know:

• Triangular Numbers ($T_n$): $\frac{n(n+1)}{2}$
• Square Numbers ($S_n$): $n^2$
• Pentagonal Numbers ($P_n$): $\frac{n(3n-1)}{2}$
• Hexagonal Numbers ($H_n$): $\frac{n(4n-2)}{2}$

Also useful: The sum of two consecutive triangular numbers, $T_{n-1} + T_n$, is a square number, $n^2$.