Class XI Chapter 4 – Principle of Mathematical Induction Maths ______________________________________________________________________________ We shall now prove that Pk 1 is true. Consider k k k k 1.3 3.5 5.7 ..... 2 1 2 1 1 1 2 1 1 2 k 4k 6k 1 2k 2 12k 2 1 Using i 3 2 k 4k 6k 1 3 2k 12k 3 2 k 4k 6k 1 3 2 4k 8k 3 2 2 4 6 1 34 8 3 k k k k k 3 3 2 2 4k 6k k 12k 24k 9 3 3 2 4k 18k 23k 9 3 3 2 2 4k 14k 9k 4k 14k 9 3 2 2 4 14 9 14 14 9 k k k k k 3 k 14k 2 14k 9 3 k 14k 2 8k 4 6k 6 1 3 2 k k k k 1 4 2 1 6 1 1 2 k k k Thus, 1 4 1 6 1 1 P k 1 3 3 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.

Class XI Chapter 4 – Principle of Mathematical Induction Maths ______________________________________________________________________________ Question 8: Prove the following by using the principle of mathematical induction for all n N: 2 2 n n 1.2 2.2 3.2 .... n.2 n1 2 1 2 Solution 8: Let the given statement be P(n), i.e., P n n n 2 2 n n1 :1.2 2.2 3.2 .... .2 1 2 2 For n = 1, we have **11** P 1 :1.2 2 **11** 2 2 0 2 2 , which is true. Let P(k) be true for some positive integer k, i.e., k 2 2 k 1 1.2 2.2 3.2 .... .2 1 2 2..... We shall now prove that Consider k k i P k 1 is true. 1.2 2.2 3.2 ..... k.2 k1 .2 2 2 k k1 k k k 1 2 2 1 2 1 k1 1 k k k 2 1 1 2 k 1 2 .2k 2 k1 1 k.2 2 k 1 1 k 1 1 2 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 9: Prove the following by using the principle of mathematical induction for all 1 1 1 1 1 ...... 1 n n 2 4 8 2 2 Solution 9: Let the given statement be P(n), i.e., 1 1 1 1 1 Pn : ...... 1 n n 2 4 8 2 2 For n 1, we have n N:

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