The Nth triangle number is given by N(N+1)/2.

The first number is

The first difference is

The second differences are all

The polynomial describing this series can be written as

= 1 + (n-1)((4 + n - 2)/2)

= 1 + (n-1)(n+2)/2

= (2 + n^2 - n + 2n - 2)/2

= (n^2 + n)/2

= n(n+1)/2

The first number is

**1**.The first difference is

**2**.The second differences are all

**1**.The polynomial describing this series can be written as

**1**+ (n-1)/1(**2**+ (n-2)/2(**1**))= 1 + (n-1)((4 + n - 2)/2)

= 1 + (n-1)(n+2)/2

= (2 + n^2 - n + 2n - 2)/2

= (n^2 + n)/2

= n(n+1)/2