NCERT Book Class 11 Maths Principle Of Mathematical Induction Questions

4.1 Overview Mathematical induction is one of the techniques which can be used to prove variety of mathematical statements which are formulated in terms of n, where n is a positive integer.

4.1.1 The principle of mathematical induction Let P(n) be a given statement involving the natural number n such that
(i) The statement is true for n = 1, i.e., P(1) is true (or true for any fixed natural number) and
(ii) If the statement is true for n = k (where k is a particular but arbitrary natural number), then the statement is also true for n = k + 1, i.e, truth of P(k) implies the truth of P(k + 1). Then P(n) is true for all natural numbers n.

4.2 Solved Examples
Short Answer Type Prove statements in Examples 1 to 5, by using the Principle of Mathematical Induction for all n ∈ N, that :

Example 1 1 + 3 + 5 + ... + (2n – 1) = n²

Solution Let the given statement P(n) be defined as P(n) : 1 + 3 + 5 +...+ (2n – 1) = n², for n ∈ N. Note that P(1) is true, since
                                         P(1) : 1 = 1²
Assume that P(k) is true for some k ∈ N, i.e., P(k) : 1 + 3 + 5 + ... + (2k – 1) = k 2 Now, to prove that P(k + 1) is true, we have
                 1 + 3 + 5 + ... + (2k – 1) + (2k + 1)
                                   = k² + (2k + 1) (Why?)
                                   = k² + 2k + 1 = (k + 1)²

Please refer to attached file for NCERT Class 11 Maths Principle Of Mathematical Induction Questions