Question

n^3 + 5n is divisible by which of the following?

(a) 3

(b) 5

(c) 7

(d) 11

The question was asked in unit test.

This is a very interesting question from The Principle of Mathematical Induction topic in chapter Principle of Mathematical Induction of Mathematics – Class 11

Answer 1

Right answer is (a) 3

To elaborate: P(n) = n^3 + 5n

P(1) = 1 + 5

P(1) = 6

We assume the P(k) is true and divisible by 6.

P(k) = k^3 + 5k is divisible by 6 and can be written as 6c or 3 x 2c

We need to prove that P(k + 1) is divisible by 6

P(k + 1) = (k + 1)^3 + 5(k + 1)

P(k + 1) = k^3 + 1 + 3k^2 + 3k + 5k + 5

P(k + 1) = (k^3 + 5k) + 3k^2 + 3k + 6

P(k + 1) = 6c + 3(k^2 + k + 2)

P(k + 1) = (3 x 2c) + 3(k^2 + k + 2)

Therefore, P(k + 1) is definitely divisible by 3