Let S(n) = 1 + 2 + + n be the sum of the

Question:

Let S(n) = 1 + 2 + · · · + n be the sum of the first n natural numbers and let C(n) = 1³ + 2³ + · · · + n³ be the sum of the first n cubes. Prove the following equalities by induction on n, to arrive at the curious conclusion that C(n) = S²(n) for every n.

a. S(n) = 1/2n(n + 1).

b. C(n) = 1/4 (n⁴ + 2n³ + n²) = 1/4n²(n + 1)².

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