Question

Prove by induction showing each step that 2+4+6+...+2n=n(n+1)

Answer 1

First you see if it holds for n=1. We see that 2(1) = (1)(1+1), so that is truly equal.

Now, we assume that it holds for n=k and try to show that it holds for n = k + 1
We add 2(k + 1) to each side
(2 + 4 + 6 + ... + 2k) + 2(k + 1) = k(k + 1) + 2(k + 1)
The right side is then:

k^2 + k + 2k + 2 = k^2 + 3k + 2 = (k + 1)(k + 2) = (k + 1)[(k + 1) + 1]

And we see that this is just n(n+1) when n = k+1 ,so this is proven now.

Answer 2

Is this making sense to you now?