1 Prove that for every natural n 1 2 n nn 1 2SolutionWe

(1) Prove that for every natural n

1 + 2 +···+ n = (n(n + 1))/ 2

Solution

We prove by induction on n.

For n = 1 we check that 1 = 1 · (1 + 1)/ 2 . Suppose that it is true for n = m. Then

1 + 2 + · · · + m + (m + 1) = (1 + 2 + · · · + m) + (m + 1) = = m(m + 1)/ 2 + (m + 1) = (m + 1)(m + 2)/ 2 . So it is true for n = m + 1. Now it is true for all positive integers n by the induction principle.

hence proved