Let S(n) = 1 + 2 + + n be the sum of the first n natural numbers and let C(n) = 13 + 23 + + n 3 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) = S 2 (n) for every n.

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

2. C(n) = 1/4 (n 4 + 2n 3 + n 2 ) = 1/4 n^2 (n + 1)^2 .

Only proof " part 1" by induction method