Question

# Prove the following by using the principle of mathematical induction for all n∈N:12+32+52+.......+(2n−1)2=n(2n−1)(2n+1)3

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Solution

## Let the given statement be P(n) i.e.,12+32+52+.......+(2n−1)2=n(2n−1)(2n+1)3For n=1 we haveP(1)=12=1=1(2.1−1)(2.1+1)3=1.1.33=1, which is true.Let P(k) be true for some k∈N i.e.,12+32+52+.......+(2n−1)2=k(2k−1)(2k+1)3.....(i)We shall now prove that P(k+1) is true.Consider12+32+52+.......+(2k−1)2+{2(k+1)−1}2 [Using (i)]=k(2k−1)(2k+1)3+(2k+2−1)2=k(2k−1)(2k+1)3+(2k+1)2=k(2k−1)(2k+1)+3(2k+1)23=(2k+1){k(2k−1)+3(2k+1)}3=(2k+1)(2k2−k+6k+3)3=(2k+1)(2k2+5k+3)3=(2k+1)(2k2+2k+3k+3)3=(2k+1){2k(k+1)+3(k+1)}3=(2k+1)(k+1)(2k+3)3=(k+1){2(k+1)−1}{2(k+1)+1}3Thus 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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