Class XI Chapter 4 – Principle of Mathematical Induction Maths ______________________________________________________________________________ 2 1 3 k 4k1 3k1 3k4 2 1 3 k 3k k 1 3k1 3k4 3k1 k1 3k13k4 1 k 3 k 1 1 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. Question 17: Prove the following by using the principle of mathematical induction for all 1 1 1 1 n ...... 3.5 5.7 7.9 2 1 2 3 3 2 3 Solution 17: Let the given statement be P(n), i.e., n n n 1 1 1 1 n Pn : ...... 3.5 5.7 7.9 2 1 2 3 3 2 3 For n = 1, we have 1 1 1 P 1: 3.5 3 2.1 3 3.5 n n n , which is true. Let P(k) be true for some positive integer k, i.e., 1 1 1 1 k Pk : .... .... 1 3.5 5.7 7.9 2k 1 2k 3 3 2k 3 We shall now prove that Pk 1 is true. Consider 1 1 1 1 1 .... 3.5 5.7 7.9 2k12k3 2k1 1 2k1 3 k 1 [Using (1)] 3 2k 3 2k 3 2k 5 n N:

Class XI Chapter 4 – Principle of Mathematical Induction Maths ______________________________________________________________________________ 1 k 1 2k3 3 2k5 1 k 2k5 3 2k3 3 2k5 2 1 2k 5k3 2k3 32k5 2 1 2k 2k 3k 3 2k3 3 2k5 1 2k k 1 3 k 1 2k3 3 2k5 k12k3 k k 3 2 3 2 5 k 1 3 2 k 1 3 Thus, Pk 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. Question 18: Prove the following by using the principle of mathematical induction for all 1 2 1 2 3 .... n 2n1 8 Solution 18: Let the given statement be P(n), i.e., P n n 1 n 8 :1 2 3 .... 2 1 2 It can be noted that P(n) is true for n = 1 since 1 1 2.**11** 2 9 8 8 Let P(k) be true for some positive integer k, i.e., 1 2 1 2 .... k 2k 1 ...... 1 8 n N:

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