Class XI Chapter 4 – Principle of Mathematical Induction Maths ______________________________________________________________________________ Thus, Pk 1 is true whenever P(k) is true. Hence, by the principle of mathematical induction, statement P(n) is true for all numbers i.e., N. Question 23: Prove the following by using the principle of mathematical induction for all n n 41 14 is a multiple of 27. Solution 23: Let the given statement be :41 14 P n n It can be observed that n Pn is a multiple of 27. Pn , i.e., is true for n = 1 Since , which is a multiple of 27. Let P(k) be true for some positive integer k, i.e., k 41 14 1 1 41 14 27 k is a multiple of 27 k k 41 14 27 m, We shall now prove that Consider 41 14 k1 k1 k k 41 .4**11**4 .14 m N...... 1 P k 1 k k k k 41 41 14 14 14 14 k 41.27 m 14 4**11**4 41.27m 27.14 k 27 41m 14 k 27 r, where Therefore, Thus, P k 1 r 41 14 k1 k1 41m14 k is true whenever P(k) is true. is a natural number. is a multiple of 27. is true whenever P(k) is true. n N: Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.

Class XI Chapter 4 – Principle of Mathematical Induction Maths ______________________________________________________________________________ Question 24: Prove the following by using the principle of mathematical induction for all n N: 2n 7 n 3 2 Solution 24: Let the given statement be : 2 7 3 2 P n n n It can be observed that Since Let 2 Pn 2.1 7 9 1 3 16 Pk Pn, i.e., is true for n = 1 , which is true. be true for some positive integer k, i.e., 2 2k 7 k 3 ...... 1 We shall now prove that Consider k k 2 1 7 2 7 2 P k 1 2 is true whenever 2 k 1 7 2k 7 2 k 3 2 2 2 k 1 7 k 6k 9 2 2 2 k 1 7 k 6k **11** Now, 2 2 k k k k 6 **11** 8 16 k k 2 2 1 7 4 2 k k 2 1 7 1 3 Thus, Pk 1 is true whenever P(k) is true. Pk is true. [Using (1)] 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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