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 + … + (2*k* – 1) (2*k* + 1) + {2(*k* + 1) – 1}{2(*k* + 1) + 1} =

Taking L.H.S.,

(1.3 + 3.5 + 5.7 + … + (2*k* – 1) (2*k* + 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*.

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