# Prove the following by the principle of mathematical induction:1.3 + 3.5 + 5.7 + … + (2n – 1) (2n + 1) Let P(n): 1.3 + 3.5 + 5.7 + … + (2n – 1) (2n + 1) For n = 1

P(1): (2.1 – 1) (2.1 + 1) = = 1x3 = = 3 = 3

Since, P(n) is true for n =1

Now, For n = k, So

1.3 + 3.5 + 5.7 + … + (2k – 1) (2k + 1) - - - - - - - (1)

We have to show that,

1.3 + 3.5 + 5.7 + … + (2k – 1) (2k + 1) + (2k + 1)(2k + 3) Now,

1.3 + 3.5 + 5.7 + … + (2k – 1) (2k + 1) + (2k + 1)(2k + 3)

= + (2k + 1)(2k + 3) using equation (1)

= = = = = = Therefore, P(n) is true for n=k + 1

Hence, P(n) is true for all n N by PMI

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