Class XI Chapter 4 – Principle of Mathematical Induction Maths ______________________________________________________________________________ Solution **11**: Let the given statement be Pn , i.e., 1 1 1 1 n n 3 P n : ..... 1.2.3 2.3.4 3.4.5 n n 1 n 2 4 n 1 n 2 For n = 1, we have 1 1 3 1 14 1 P1: 1 2 3 4 1 1 1 2 4 2 3 1 2 3 Let P(k) be true for some positive integer k, i.e., , which is true. kk 3 1 1 1 1 ...... 1.2.3 2.3.4 3.4.5 k k 1 k 2 4 k 1 k 2 We shall now prove that Consider P k 1 is true. …… (i) 1 1 1 1 1 ...... 123 234 345 k k 1 k 2 k 1 k 2 k 3 kk 3 1 k k k k k 4 1 2 1 2 3 1 k k 3 1 k 1 k 2 4 k 3 2 1 k k3 4 k 1k 2 4k 3 2 1 k k 6k 9 4 k 1k 2 4k 3 3 2 1 k 6k 9k 4 k 1k 2 4k 3 3 2 2 1 k 2k k 4k 8k 4 k 1k 2 4k 3 2 2 2 1 4 2 1 1 k k k k k k 1k 2 4k 3 1 4k 1 2 2 1 k k k 1k 2 4 k 3 [Using (i)]

Class XI Chapter 4 – Principle of Mathematical Induction Maths ______________________________________________________________________________ 2 k1 k4 k k k 4 1 2 3 k1 k1 3 4 k1 1 k1 2 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 12: Prove the following by using the principle of mathematical induction for all a r a ar ar ..... ar r 1 n 1 2 n1 Solution 12: Let the given statement be P(n), i.e., n a r 1 2 n1 P n : a ar ar ..... ar For n = 1, we have 1 a r 1 P1: a a r 1 , which is true. r 1 Let P(k) be true for some positive integer k, i.e., k a r 1 2 k1 a ar ar ...... ar ....... i r 1 We shall now prove that Pk 1 is true. Consider 2 k 1 k ..... k ar 1 1 1 a ar ar ar ar i k ar Using r 1 k k 1 1 a r ar r r 1 1 1 a r ar ar r 1 k k k n N:

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