For each positive integer n, let P(n) be the following inequality.
2^n < (n + 1)!
(a) What is P(2)?
4 < 4
4 < 2
4 < 6
2^n < (n + 2)!
4 < (n + 2)!
Is P(2) true?
Yes
No
(b) What is P(k)?
2^k < k!
2^n < (n + k)!
k^2 < (k + 1)!
k^2 < k!
2^k < (k + 1)!
(c) What is P(k + 1)?
2^n + 1 < (n + k + 1)!
2^{k + 1} < (k + 1)!
(k + 1)^2 < (k + 2)!
2^{k + 1} < (k + 2)!
(k + 1)^2 < (k + 1)!
(d) In a proof by mathematical induction that this inequality holds for every integer n ≥ 2, what must be shown in the inductive step?
We need to show that if k is any integer with k ≥ 2 and if P(k) is true, then P(k + 1) is also true.
We need to show that if k is any integer with k ≥ 2 and if P(k) is true, then P(k + 1) is false.
We need to show that if k is any integer with k ≥ 2 and if P(k) is false, then P(k + 1) is true.
We need to show that if k is any integer with k ≥ 2 and if P(k) is false, then P(k + 1) is also false.