2 + 4 + 6 + …+ 2n = n² + n for all natural numbers n.

Given; P(n) is 2 + 4 + 6 + …+ 2n = n² + n.

P(0) = 0 = 0² + 0 ; it’s true.

P(1) = 2 = 1² + 1 ; it’s true.

P(2) = 2 + 4 = 2² + 2 ; it’s true.

P(3) = 2 + 4 + 6 = 3² + 2 ; it’s true.

Let P(k) be 2 + 4 + 6 + …+ 2k = k² + k is true;

⇒ P(k+1) is 2 + 4 + 6 + …+ 2k + 2(k+1) = k² + k + 2k +2

= (k² + 2k +1) + (k+1)

= (k + 1)² + (k + 1)

⇒ P(k+1) is true when P(k) is true.

∴ By Mathematical Induction 2 + 4 + 6 + …+ 2n = n² + n is true for all natural numbers n.