Answer to Question #198634 in Calculus for Moe

Question

3) Prove by induction that for all natural numbers n:

(e) n² + n is divisible by 2. (Can you prove this directly?)

Answer 1

Induction on positive integers $n :$

When $n =1, n^2+n = 2$ which is obviously divisible by $2.$

Assume when $n = k$ , that $k^2+k$ is divisible by $2$ .

When $n = k+1$ , we have

$(k+1)^2+k+1 = k^2+2k+1+k+2$

$= (k^2+k)+2(k+1)$

From our assumption, $k^2+k$ is divisible by $2$ and so the whole expression is divisible by 2.

Therefore, by induction $n^2 + n$ is divisible by 2 for all natural numbers n.