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
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.
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.
