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