Class XI Chapter 4 – Principle of Mathematical Induction Maths ______________________________________________________________________________ Question 2: Prove the following by using the principle of mathematical induction for all n N: 2 3 3 3 3 n n 1 1 2 3 ..... n 2 Solution 2: Let the given statement be Pn , i.e., 2 3 3 3 3 n n1 P n :1 2 3 ...... n 2 For P Let n 1, we have 2 2 1 **11** 1.2 2 2 3 2 1 :1 1 1 1 Pk , which is true. be true for some positive integer k, i.e., 3 3 3 3 k k1 1 2 3 ..... k ....... 2 We shall now prove that Consider P k 1 k 3 3 3 3 3 1 2 3 ...... k 1 1 3 2 3 3 3 ..... k 3 k 1 3 2 2 is true. i k k1 3 k1 Using i 2 2 k k1 k 1 4 2 1 4 1 2 k k 2 k 3 4 2 2 k 1 k 4k 1 4 k 1 2 k 2 4k 4 4 k1 k2 2 2 4 3

Class XI Chapter 4 – Principle of Mathematical Induction Maths ______________________________________________________________________________ k1 k**11** Thus, 2 2 4 2 k1 k**11** 2 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 3: Prove the following by using the principle of mathematical induction for all 1 1 1 2n 1 ..... 1 2 1 2 3 1 2 3 ... n n1 Solution 3: Let the given statement be P n For , P n i.e., 1 1 1 2n :1 ..... 1 2 1 2 3 1 2 3 .... n n1 n 1, we have 2.1 2 P 1 :1 1, **11** 2 Let Pk which is true. be true for some positive integer k, i.e., 1 1 1 2k 1 ... .... ....... i 1 2 1 2 3 1 2 3 ..... k k1 We shall now prove that Pk 1 is true. Consider 1 1 1 1 1 ..... 1 2 1 2 3 1 2 3 ..... k 1 2 3 ..... k k 1 1 1 1 1 1 .... 1 2 1 2 3 1 2 3 ..... k 1 2 3 .... k k 1 2k 1 k 1 1 2 3 ..... k k 1 [Using (i)] n N:

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