Question

What would be the hypothesis of mathematical induction for n(n + 1) < n! (where n ≥ 4) ?

(a) It is assumed that at n = k, k(k + 1)! > k!

(b) It is assumed that at n = k, k(k + 1)! < k!

(c) It is assumed that at n = k, k(k + 1)! > (k + 1)!

(d) It is assumed that at n = k, (k + 1)(k + 2)! < k!

The question was posed to me in a national level competition.

The query is from The Principle of Mathematical Induction topic in portion Principle of Mathematical Induction of Mathematics – Class 11

Answer 1

Correct option is (b) It is assumed that at n = k, k(k + 1)! < k!

For explanation I would say: When we use the principle of mathematical induction, we assume that P(n) is true for P(k) and prove that P(k + 1) is also true. Here P(k) is k(k + 1)! < k!