Question

For all n ∈ N, by using principle of mathematical induction, then prove that:

1.3 + 3.5 + 5.7 + .... + (2n - 1)(2n + 1) = (n(4n² + 6n - 1))/3

Answer 1

Let the given statement is P(n), where n ∈ N

P(n) = 1.3 + 3.5 + 5.7 +……+ (2n – 1)(2n + 1)

Hence, given statement is true for n = 1, 

i.e, P(1) is true

Let given statement is true n = m, 

Hence, given statement is also true for n = m + 1

i.e., P (m + 1) is true.

Hence, by the principle of mathematical induction we can say that the given statement is true for each natural number n ∈ N.

Hence Proved.