CLASS_11_MATHS_SOLUTIONS_NCERT

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Class XI Chapter 4 – Principle of Mathematical Induction Maths ______________________________________________________________________________ k k k 1.3 2.3 3.3 ..... .3 1 .3 k 1 2k 1 3 3 2 3 k1 k 1 k 13 Using i 4 2k1 3 3 4 k1 3 k1 k1 4 1 k k k 3 2 1 4 1 3 4 k 3 6 3 3 4 1 k k 1 3 .3 2k 1 3 k1 1 4 3 2k 1 3 4 Thus, k 1 1 2 k 1 1 3 3 4 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 6: Prove the following by using the principle of mathematical induction for all n n 1 n 2 1.2 2.3 3.4 .... n. n1 3 Solution 6: Let the given statement be P(n), i.e., 1n 2 n n P n : 1.2 2.3 3.4 .... n. n 1 3 For n = 1, we have 1111 2 1.2.3 P1 :1.2 2 2 , which is true. 3 3 Let P(k) be true for some positive integer k, i.e., n N:
Class XI Chapter 4 – Principle of Mathematical Induction Maths ______________________________________________________________________________ 1k 2 k k 1.2 2.3 3.4 ..... k. k 1 ....... i 3 We shall now prove that Consider P k 1 is true. 1.2 2.3 3.4 .... k. k 1 k 1 . k 2 1.2 2.3 3.4 ..... k. k 1 k 1 . k 2 1k 2 k k k 1k 2 Using i 3 k k1k 2 1 3 k 1k 2k 3 3 Thus, k 1 k 11 k 1 2 P k 1 3 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 7: Prove the following by using the principle of mathematical induction for all n n n 1.3 3.5 5.7 ..... 2n12n1 3 Solution 7: Let the given statement be P(n), i.e., 2 4 6 1 n n n Pn :1.3 3.5 5.7 ..... 2n 12n 1 3 For n = 1, we have 2 1 4.1 6.11 4 6 1 9 P 1 :1.3 3 3 3 3 3 Let Pk be true for some positive integer k, i.e., 2 4 6 1 , which is true. 2 k 4k 6k 1 1.3 3.5 5.7 ..... 2k 12k 1 ....... i 3 n N:
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Class XI Chapter 1 - Sets Maths ___
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Class XI Chapter 2 - Relations and
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Class XI Chapter 6 - Linear Inequal
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Class XI Chapter 7 - Permutation an
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Class XI Chapter 8 - Binomial Theor
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Class XI Chapter 9 - Sequences and
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Class XI Chapter 10 - Straight Line
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Class XI Chapter 11 - Conic Section
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Class XI Chapter 12 - Introduction
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Class XI Chapter 13 - Limits and De
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Class XI Chapter 14 - Mathematical
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Class XI Chapter 15 - Statistics Ma
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