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Ex 4.1, 15 - Prove 12 + 32 + 52 ..+ (2n-1)2 - Chapter 4 Induction - Ex 4.1

Ex 4.1, 15 - Chapter 4 Class 11 Mathematical Induction - Part 2
Ex 4.1, 15 - Chapter 4 Class 11 Mathematical Induction - Part 3 Ex 4.1, 15 - Chapter 4 Class 11 Mathematical Induction - Part 4

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Ex 4.1,15 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 Let P (n) : 12 + 32 + 52 + ..+(2n 1)2 = (n(2n 1)(2n + 1))/3 For n = 1, L.H.S = 12 = 1 R.H.S = (1(2 1 1)(2 1+ 1))/3 = (1(2 1) (2 + 1))/3 = (1 1 3)/3 = 1 Hence L.H.S. = R.H.S P(n) is true for n = 1 Assume that P(k) is true 12 + 32 + 52 + ..+ (2k 1)2 = (k(2k 1)(2k + 1))/3 We will prove that P(k + 1) is true. 12 + 32 + 52 + + (2(k + 1) 1)2 = ( (k + 1)(2(k + 1) 1)(2(k + 1)+ 1))/3 12 + 32 + 52 + + (2k 1)2 + (2k + 2 1)2 = ((k + 1)(2k + 2 1)(2k + 2 + 1))/3 12 + 32 + 52 + + (2k 1)2 + (2k + 1)2 = ((k + 1)(2k + 1)(2k + 3))/3 From (1) 12 + 32 + 52 + ..+ (2k 1)2 = (k(2k 1)(2k + 1))/3 Adding (2k + 1)2 both sides 12 + 32 + 52 + ..+ (2k 1)2 + (2k + 1)2 = (k(2k 1)(2k + 1))/3 + (2k + 1)2 = (k(2k 1)(2k + 1) + 3(2 + 1)^2)/3 = (2k + 1)((k(2k 1)+ 3(2 + 1))/3) = (2k + 1)((k(2k) k(1)+ 3(2 ) +3(1))/3) = (2k + 1)((2k^2 + 6 + 3)/3) = (2k + 1)((2k^2 + 5 + 3)/3) = (2k + 1)((2k^2 + 2 + 3 + 3)/3) = (2k + 1)((2k(k + 1) + 3( +1))/3) = (2k + 1)(((2k + 3)( +1))/3) = ((k + 1)(2k + 1)(2k + 3))/3 Thus, 12 + 32 + 52 + + (2k 1)2 + (2k + 1)2 = ((k + 1)(2k + 1)(2k + 3))/3 which is the same as P(k + 1) P(k+1) is true when P(k) is true By the principle of mathematical induction, P(n) is true for n, where n is a natural number

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.