Question

Prove the following statement by mathematical induction. For every integer n ≥ 0, 2n <(n + 2)!

Proof (by mathematical induction): Let P(n) be the inequality 2n < (n + 2)!.

We will show that P(n) is true for every integer n ≥ 0. Show that P(0) is true: Before simplifying, the left-hand side of P(0) is _______ and the right-hand side is ______ . The fact that the statement is true can be deduced from that fact that 20 = 1. Show that for each integer k ≥ 0, if P(k) is true, then P(k + 1) is true: Let k be any integer with k ≥ 0, and suppose that P(k) is true. In other words, suppose that ______. [This is P(k), the inductive hypothesis.] We must show that P(k + 1) is true. P(k + 1) is the inequality ______ . Information about P(k + 1) can be deduced from the following steps. Identify the reason for each step.

Please circle the answers.

Answer 1