Prove the following by the principle of mathematical induction: 1 2 + 3 2 + 5 2 + … + (2n – 1) 2

Question

Philoid · Accepted Answer

Let P(n): 1² + 3² + 5² + … + (2n – 1)² =

For n = 1

= (2.1 – 1)² =

= 1 = 1

Since, P(n) is true for n = 1

Let P(n) is true for n = k ,

P(k) ): 1² + 3² + 5² + … + (2k – 1)² = - - (1)

We have to show that,

1² + 3² + 5² + … + (2k – 1)² + (2k + 1)² =

Now,

{1² + 3² + 5² + … + (2k – 1)²} + (2k + 1)²

= using equation (1)

=

= (2k + 1)

=

Therefore, P(n)is true for n = k + 1

Hence, P(n) is true for all n∈ N