Question

Using the principle of mathematical induction, prove each of the following for all n ϵ N: 

1 + 2 + 3 + 4 + … + n = 1/2 n(n + 1)

Answer 1

To Prove: 

1 + 2 + 3 + 4 + … + n = 1/2 n(n + 1) 

Steps to prove by mathematical induction: 

Let P(n) be a statement involving the natural number n such that 

(i) P(1) is true 

(ii) P(k + 1) is true, whenever P(k) is true 

Then P(n) is true for all n ϵ N 

Therefore, 

Let P(n): 1 + 2 + 3 + 4 + … + n = 1/2 n(n + 1) 

Step 1: 

P(1) = 1/2 1(1 + 1) = 1/2 × 2 = 1 

Therefore, P(1) is true 

Step 2: 

Let P(k) is true Then, 

P(k): 1 + 2 + 3 + 4 + … + k = 1/2 k(k + 1) 

Now, 

1 + 2 + 3 + 4 + … + k + (k + 1) = 1/2 k(k + 1) + (k + 1) 

= (k + 1){ 1/2 k + 1} 

= 1/2 (k + 1) (k + 2) 

= P(k + 1) 

Hence, P(k + 1) is true whenever P(k) is true 

Hence, by the principle of mathematical induction, we have 

1 + 2 + 3 + 4 + … + n = 1/2 n(n + 1) for all n ϵ N 

Hence proved.