Figurate numbers: Triangular, Pentagonal Polygonal numbers

Triangular numbers

The $n$th triangular number is

$T_n =1+2+3+ ··· + n =  \dfrac{1}{2}n(n + 1)$.

The first few of these are 

$1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ... .$

Pentagonal numbers

The pentagonal numbers are the sums of the arithmetic progression

$1+4+7+ ··· + (3n − 2) + ··· $.

The nth pentagonal number is

$P_n = \dfrac{1}{2}n(3n − 1)$.

The polygonal numbers $P_{n,k}$

More generally, for a fixed $k$, the $k$-gonal numbers are the sums of the arithmetic progression 

$1+(k − 1) + (2k − 3) + ···$. 

The nth $k$-gonal number is 

$P_{k,n} = \dfrac{1}{2}n((k − 2)n − (k − 4))$.