 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2008/paper-1/10/ii/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

The identity

(n + 1)^{3} - n^{3} = 3 n (n + 1) + 1

, summed from zero to

N

, gives

(N + 1)^{3} = 3 \sum_{n = 0}^{N} n (n + 1) + (N + 1)

. Hence

n = 0 \sum N n (n + 1) = \frac{N ( N + 1 ) ( N + 2 )}{3} .

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