Calculus polynomial

Question

Use polynomial fitting to find a closed formula for the sequence −2,0,−4,−14,−30,−52,...

Answer 1

S = -2, 0, -4, -14, -30, -52

Differences of each successive term in S = 2, -4, -10, -16, -22

The differences can be seen to take on the closed form solution D(n) = 8 - 6*n, where the first term in the sequence is n=1.

Now we can find a solution for S:

S(n) = -2 + ∑_i=1^n-1 D(i) = -2 + (n-1)*(8) - 6*∑_i=1^n-1i = -2 + (n-1)*(8) - 6*(n-1)(n)/2 = -2 + (n-1)(8 - 6n/2)

Note: ∑_i=1ⁿ i = n*(n+1)/2