Class XI Chapter 4 – Principle of Mathematical Induction Maths ______________________________________________________________________________ ar a ar ar r 1 k k1 k k 1 ar a r 1 k 1 a r 1 r 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 13: Prove the following by using the principle of mathematical induction for all 3 5 7 2n 1 1 1 1 ..... 1 n 1 2 1 4 9 n Solution 13: Let the given statement be P(n), i.e., 2 3 5 7 2n 1 P n : 1 1 1 ..... 1 1 2 1 4 9 n For n = 1, we have 3 2 2 P 1 : 1 4 **11** 2 4 1 , which is true. Let P(k) be true for some positive integer k, i.e., n 2 3 5 7 2k 1 2 1 1 1 ..... 1 k 1 ...... 1 2 1 4 9 k P k 1 We shall now prove that is true. Consider 3 5 7 2k 1 2k 1 1 1 1 1 .... 1 1 1 4 9 k 2 k 1 2 2 2k 1 1 k 1 1 2 k 1 2 2 k1 2k1 1 k 1 2 k 1 2 k k 1 2 1 1 Using 1 n N:

Class XI Chapter 4 – Principle of Mathematical Induction Maths ______________________________________________________________________________ 2 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 14: Prove the following by using principle of mathematical induction for all 1 1 1 1 1 1 1 ..... 1 n 1 1 2 3 n Solution 14: Let the given statement be P(n), i.e., 1 1 1 1 P n : 1 1 1 ..... 1 n 1 1 2 3 n For n = 1, we have P 1 1 1 : 1 2 **11** , which is true. Let P(k) be true for some positive integer k, i.e., 1 1 1 1 1 2 3 k : 1 1 1 .... 1 k 1 ....... 1 P k We shall now prove that Consider P k 1 is true. 1 1 1 1 1 1 1 1 ..... 1 1 1 2 3 k k1 1 k 1 1 k 1 1 1 k 1 k k 1 k 1 1 Thus, P k 1 is true whenever P(k) is true. Using 1 n N: Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.

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Class XI Chapter 12 - Introduction

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