Class XI Chapter 4 – Principle of Mathematical Induction Maths ______________________________________________________________________________ Question 15: Prove the following by using the principle of mathematical induction for all n N: n n n 1 3 5 ..... 2n 1 3 2 2 2 2 2 1 2 1 Solution 15: Let the given statement be P(n), i.e., n n n P n :1 3 5 ..... 2n 1 3 2 2 2 2 2 1 2 1 For n 1, we have 2 1 2.1 1 2.1 1 1.1.3 P1 1 1 1, 3 3 Let P(k) be true for some positive integer k, i.e., k P k which is true. 2 2 2 2 k 2k 1 2k 1 1 3 5 .... 2 1 ...... 1 3 We shall now prove that Consider P k 1 is true. 2 2 2 k 2 k 2 1 3 5 ..... 2 1 2 1 1 2 12 1 k k k 3 2 12 1 k k k 3 2k 2 1 2 2k 1 2 k k k 2 2 2 1 2 1 3 2 1 3 2k 1 k 2k 1 3 2k 1 2k 12k 2 k 6k 3 3 3 2k 12k 2 5k 3 3 2k 12k 2 2k 3k 3 3 2k 1 2k k 1 3k 1 3 Using 1

Class XI Chapter 4 – Principle of Mathematical Induction Maths ______________________________________________________________________________ 2k 1k 12k 3 3 k 1 2 k 1 1 2 k 1 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 16: Prove the following by using the principle of mathematical induction for all 1 1 1 1 n .... 1.4 4.7 7.10 3 2 3 1 3 1 n n n Solution 16: Let the given statement be P(n), i.e., 1 1 1 1 n Pn: .... 1.4 4.7 7.10 3n 2 3n 1 3n 1 For n = 1, we have 1 1 1 1 P 1 , 1.4 3.**11** 4 1.4 which is true. Let P(k) be true for some positive integer k, i.e., 1 1 1 1 k Pk ..... ...... 1 1.4 4.7 7.10 3k 2 3k 1 3k 1 We shall now prove that Pk 1 is true. Consider 1 1 1 1 1 .... 1.4 4.7 7.10 3k23k1 3k1 23 k1 1 k 1 3k 1 3k 1 3k 4 1 k 1 3k1 3k4 1 k 3k4 1 3k1 3k4 [Using (1)] n N:

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