 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/i/solution

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

The two finite summations cancel term by term after shifting the index. Only the final positive term remains:

n = 0 \sum N (n + 1)^{3} - n = 0 \sum N n^{3} = (N + 1)^{3} .

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