Prove the following by the principle of mathematical induction: `\ 1. 3+2. 4+3. 5++(2n-1)(2n+1)=(n(4n^2+6n-1))/3`

Explanation

Given
` P(n) : 1.3 + 3.5 + 5.7 + ........... + (2n -1)(2n+1)
=(n*(4n^2+6n−1))​/3`

Putting n=1,

L.H.S.=R.H.S=3
so, P(n) is true for n = 1

Assuming that P(n) is true for n = k

`P(k) : 1.3 + 3.5 + 5.7 + ............ + (2k-1)(2k+1)
=k(4k^2+6k−1)/3​`

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