We define the first difference of a function by (x) (x 1) (x) Calculate (x 3 ), (x 2 ), and (x) Then find a polynomial P of degree 3 such that P (x 1) 2 and P(0) 0 Conclude that P(n) 1 2 2 2 n 2

Question: We define the first difference of a function by (x) = (x + 1) (x). Calculate (x 3 ), (x

We define the first difference δ ƒ of a function ƒ by δ ƒ(x) = ƒ(x + 1) − ƒ(x).

Calculate δ(x³), δ(x²), and δ(x). Then find a polynomial P of degree 3 such that δP = (x + 1)²and P(0) = 0. Conclude that P(n) = 1² + 2² + · · · + n².

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