Prove the following

1.3 + 3.5 + 5.7 + ..... +(2n - 1) (2n + 1 ) =n( 4n square + 6n -1)/2 plz explain briefly in a simple method

The said query is:

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

We will prove it as follows:

Let the given statement be P(n), i.e.,

P(n): …(1)

For n = 1, we have

, which is true.

Let P(k) be true for some positive integer k.

Then we must have,

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

That is, we have to prove that 

(1.3 + 3.5 + 5.7 + … + (2k – 1) (2k + 1) + {2(k + 1) – 1}{2(k + 1) + 1} =

Taking L.H.S.,

(1.3 + 3.5 + 5.7 + … + (2k – 1) (2k + 1) + {2(k + 1) – 1}{2(k + 1) + 1}

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

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