Q. 16

# Prove the following by the principle of mathematical induction:12 + 32 + 52 + … + (2n – 1)2 Let P(n): 12 + 32 + 52 + … + (2n – 1)2 = For n = 1

= (2.1 – 1)2 = = 1 = 1

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

Let P(n) is true for n = k ,

P(k) ): 12 + 32 + 52 + … + (2k – 1)2 = - - (1)

We have to show that,

12 + 32 + 52 + … + (2k – 1)2 + (2k + 1)2 = Now,

{12 + 32 + 52 + … + (2k – 1)2} + (2k + 1)2

= using equation (1)

= = (2k + 1) = (2k + 1) = (2k + 1) = = = = = = Therefore, P(n)is true for n = k + 1

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

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