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Prove the following by using principle of mathematical induction for all nN:1.3+3.5+5.7+.......+(2n1)(2n+1)=n(4n2+6n1)3

Solution
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Let the given statement be P(n), i.e.,
P(n):1.3+3.5+5.7+.......+(2n1)(2n+1)=n(4n2+6n1)3
For n=1, we have
P(1):1.3=3=1(4.12+6.11)3=4+6+13=93=3, which is true
Let P(k) be true for some kN, i.e.,
1.3+3.5+5.7+......+(2k1)(2k+1)=k(4k2+6k1)3.........(i)
We shall now prove that P(k+1) is true. i.e.1.3+3.5+5.7+.......+(2k1)(2k+1)+{2(k+1)1}{2(k+1)+1}=(k+1){4(k+1)2+6(k+1)1}3
Consider LHS,
(1.3+3.5+5.7+.......+(2k1)(2k+1)+{2(k+1)1}{2(k+1)+1}
=k(4k2+6k1)3+(2k+21)(2k+2+1) [Using (i)]
=k(4k2+6k1)3+(2k+1)(2k+3)
=k(4k2+6k1)3+(4k2+8k+3)
=k(4k2+6k1)+3(4k2+8k+3)3
=4k3+6k2k+12k2+24k+93
=4k3+18k2+23k+93
=4k3+14k2+9k+4k2+14k+93
=(k+1)(4k2+14k+9)3
=(k+1){4k2+8k+4+6k+61}3
=(k+1){4(k2+2k+1)+6(k+1)1}3
=(k+1){4(k+1)2+6(k+1)1}3
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

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