Question

By principle of mathematical induction, 2^4n-1 is divisible by which of the following?

(a) 8

(b) 3

(c) 5

(d) 7

I have been asked this question in unit test.

Enquiry is from The Principle of Mathematical Induction in section Principle of Mathematical Induction of Mathematics – Class 11

Answer 1

The correct option is (a) 8

Best explanation: P(n) = 2^4n – 1

P(1) = 2^3 = 8

Let us assume P(k) is divisible by 8 and can be written as 8c, where c is any integer.

P(k) = 2^4k – 1 = 8c

P(k + 1) = 2^4(k + 1) – 1

P(k + 1) = 2^4k + 3

P(k + 1) = 2^4 . 2^4k – 1

P(k + 1) = 2^4 . 8c

Clearly, P(k + 1) is divisible by 2, 4, 8 and 16.