Question

If P(k) = k^2 (k + 3) (k^2 – 1) is true, then what is P(k + 1)?

(a) (k + 1)^2 (k + 3) (k^2 – 1)

(b) (k + 1)^2 (k + 4) (k^2 – 1)

(c) (k + 1)^2 (k + 4) k (k + 2)

(d) (k + 1) (k + 4) k (k +2)

I had been asked this question in unit test.

Asked question is from The Principle of Mathematical Induction in portion Principle of Mathematical Induction of Mathematics – Class 11

Answer 1

Correct choice is (c) (k + 1)^2 (k + 4) k (k + 2)

Easiest explanation: In mathematical induction, if P(k) is true, we need to prove that P(k + 1) is also true. Here P(k + 1) is found by substituting (k + 1) in place of k. P(k + 1) = (k + 1)^2 (k + 1 + 3) ((k + 1)^2 – 1)

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

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

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