Question

Prove that the sum of the first n non-zero even numbers is n² + n

Answer 1

Let P(n): 2 + 4 + 6 + … + 2n = n² + n, ∀ n ∈ N

Step 1:

P(1): 2 = 1² + 1 = 2

Which is true for P(1)

Step 2:

P(k): 2 + 4 + 6 + … + 2k

= k² + k.

Let it be true.

Step 3:

P(k + 1): 2 + 4 + 6 + … + 2k + (2k + 2)

= k²+ k + (2k + 2)

= k² + 3k + 2

= k² + 2k + k + 1 + 1

= (k+ 1)² + (k + 1)

Which is true for P(k + 1)

So, P(k + 1) is true whenever P(k) is true.