The first differences are

2, 3, 4, 5, 6

The second differences are

1, 1, 1, 1

The Nth term is

f(N) = (first series term) + (N-1)/1*(first first difference) + (N-1)(N-2)/(1*2)*(first second difference)

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

= 1 + 2N - 2 + N^2/2 -3N/2 + 2/2

This is the formula for the Nth triangle number.

This method will find a polynomial that fits any series of numbers. If the third differences are non-zero, the next term of the polynomial is (N-1)(N-2)(N-3)/(1*2*3)*(first third difference). Perhaps you can begin to see the pattern.

2, 3, 4, 5, 6

The second differences are

1, 1, 1, 1

The Nth term is

f(N) = (first series term) + (N-1)/1*(first first difference) + (N-1)(N-2)/(1*2)*(first second difference)

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

= 1 + 2N - 2 + N^2/2 -3N/2 + 2/2

**f(N) = N(N+1)/2**This is the formula for the Nth triangle number.

This method will find a polynomial that fits any series of numbers. If the third differences are non-zero, the next term of the polynomial is (N-1)(N-2)(N-3)/(1*2*3)*(first third difference). Perhaps you can begin to see the pattern.