Class XI Chapter 4 – Principle of Mathematical Induction Maths ______________________________________________________________________________ 2k 1 n n1 1 2 3.... n k 1 k1k**11** 2 2 2k 2 k 1 k 1k 2 2 1 k k1 k2 2 2 k 2k1 k1 k2 2 2 k 1 k 1 k 2 k 2 k 1 2 Thus, P k 1 is true whenever Pk is true. Hence, by the principle of mathematical induction, statement numbers i.e., N. Pn is true for all natural Question 4: Prove the following by using the principle of mathematical induction for all 1 2 3 n n n n 1.2.3 2.3.4 .... nn 1n 2 4 Solution 4: Let the given statement be Pn , i.e., 1 2 3 n n n n Pn :1.2.3 2.3.4 .... nn 1n 2 4 For n 1, we have 1 1 1 1 2 1 3 1.2.3.4 P1 :1.2.3 6 6 , which is true. 4 4 Let P(k) be true for some positive integer k, i.e., 1 2 3 k k k k 1.2.3 2.3.4 ..... k k 1k 2 ...... i 4 We shall now prove that Pk 1 is true. n N:

Class XI Chapter 4 – Principle of Mathematical Induction Maths ______________________________________________________________________________ Consider 1.2.3 2.3.4 ..... k k 1 k 2 k 1 k 2 k 3 1.2.3 2.3.4 ..... k k 1k 2 k 1k 2 k 3 1 2 3 k k k k k 1k 2k 3 Using i 4 k k 1 k 2 k 3 1 4 k 1k 2k 3k 4 4 Thus, k 1 k **11** k 1 2 k 1 3 P k 1 4 is true whenever Pk is true. Hence, by the principle of mathematical induction, statement numbers i.e., N. Pn is true for all natural Question 5: Prove the following by using the principle of mathematical induction for all n N: n1 n 2 3 n 2 1 3 3 1.3 2.3 3.3 ..... n.3 4 Solution 5: Let the given statement be , P n i.e., n1 n 2 3 n 2 1 3 3 Pn :1.3 2.3 3.3 .... n3 4 For n 1, we have **11** 2 2.**11** 3 3 3 3 12 P1 :1.3 3 3 4 4 4 Let Pk be true for some positive integer k, i.e., k 1 2 3 k 2k 1 3 3 1.3 2.3 3.3 ..... k3 ....... i 4 We shall now prove that Consider P k 1 1.3 2.3 3.3 ..... k.3 k 1 .3 is true. 2 3 k k 1 , which is true.

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Class XI Chapter 2 - Relations and

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Class XI Chapter 12 - Introduction

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