NCERT solution Class 11 principle of mathematical induction

NCERT Solutions

Class 11 Maths

Principle of Mathematical Induction

NCERT Questions

Ex.4.1 Q.1
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Ex.4.1 Q.24

Ex.4.1 Q.23

Prove the following by using the principle of mathematical induction for all n ∈ N:

41ⁿ – 14ⁿis a multiple of 27.

Let the given statement be P(n), i.e.,

P(n): 41ⁿ – 14ⁿ is a multiple of 27.

It can be observed that P(n) is true for n = 1

since 41¹ – 14¹ = 27, which is a multiple of 27.

Let P(k) be true for some positive integer k, i.e.,

41^k – 14^k is a multiple of 27

So, 41^k – 14^k = 27m, where m ∈ N ............. (1)

We shall now prove that P (k + 1) is true whenever P(k) is true.

Consider

41^k+1 – 14^k+1

= 41^k × 41 – 14^k× 14

= 41(41^k – 14^{k +} 14^k) - 14^{k +} 14

= 41(41^k – 14^k)+ 41 × 14^k - 14^{k +} 14

= 41 × 27m+ 14^k (41 – 14)

= 41 × 27m+ 27 × 14^k

= 27(41m+ 14^k)

= 27 × r, where r = (41m+ 14^k) is a natural number.

Therefore, 41^k+1 – 14^k+1 is a multiple of 27

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.

Principle of Mathematical Induction Class 11 Maths NCERT Chapter 4 NCERT Solutions

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