Class XI Chapter 4 – Principle of Mathematical Induction Maths ______________________________________________________________________________ 1 1 1 P 1 : 1 , 1 2 2 2 Let Pk which is true. be true for some positive integer k, i.e., 1 1 1 1 1 ..... 1 ...... 2 4 8 2 2 k k i We shall now prove that Pk 1 is true. Consider 1 1 1 1 1 ...... k 2 4 8 2 2 k1 1 1 1 Using i k k1 2 2 1 1 1 1 k 2 2 1 1 1 2 k 2 1 1 2 k Thus, 1 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 10: Prove the following by using the principle of mathematical induction for all 1 1 1 1 n ...... 2.5 5.8 8.**11** 3n 1 3n 2 6n 4 Solution 10: Let the given statement be Pn , i.e., 1 1 1 1 n Pn : ...... 2.5 5.8 8.**11** 3 1 3 2 6 4 n n n For n 1, we have 1 1 1 1 P 1 , which is true. 2.5 10 6.1 4 10 Let P(k) be true for some positive integer k, i.e., n N:

Class XI Chapter 4 – Principle of Mathematical Induction Maths ______________________________________________________________________________ 1 1 1 1 k ..... ...... 2.5 5.8 8.**11** 3k 1 3k 2 6k 4 We shall now prove that Consider P k 1 is true. i 1 1 1 1 1 ..... 2.5 5.8 8.**11** 3 1 3 2 3 1 1 3 1 2 k k k k k 1 Using i 6k 4 3k 31 3k 3 2 k 6k 4 3k 2 3k 5 1 k 1 2 3k 2 3k 2 3k 5 1 k 1 3k2 2 3k5 1 k 3k5 2 3k2 2 3k5 2 1 3k 5k2 3k2 23k5 3k2k1 k 1 3k2 2 3 5 k 1 6k 10 k 1 6 k 1 4 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 **11**: Prove the following by using the principle of mathematical induction for all n N: nn 3 1 1 1 1 ..... 1.2.3 2.3.4 3.4.5 n n 1 n 2 4 n 1 n 2

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