Prove the following by using the principle of mathematical induction for all n ∈ N

1³ + 2³ + 3³ + … + n³ = [n (n + 1)/2]²

## Solution:

Let P (n) be the given statement.

i.e., P (n) : 1³ + 2³ + 3³ + … + n³ = [n (n + 1)/2]²

For n = 1,

P (1) : 1³ = [1 (1 + 1)/2]²

1= [(1 x 2)/2]²

1 = 1, which is true.

Assume that P (k) is true for some positive integer k.

1.e., P (k) : 1³ + 2³ + 3³ + … + k³ = [k (k + 1)/2]² ….(1)

We will now prove that P (k + 1) is also true.

Now, we have

1³ + 2³ + 3³ + … + (k + 1)³

= (1³ + 2³ + 3³ + … + k³) + (k + 1)³

= [k (k + 1)/2]² + (k + 1)³ … From (1)

= [k² (k + 1)²/4] + (k + 1)³

= [k² (k + 1)² + 4 (k + 1)³]/4

= (k + 1)² [k² + 4 (k + 1)]/4

= (k + 1)² [k² + 4k + 4]/4

= [(k + 1)²(k + 2)²]/4

= [(k + 1)(k + 2)/2]²

= [(k + 1)(k + 1 + 1)/2]²

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

Hence, from the principle of mathematical induction, the statement P (n) is true for all natural numbers i.e., n ∈ N .