Question

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

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

For each positive integer n, let P(n) be the following inequality.
2^n < (n + 1)!
(a) What is P(2)?
O 4 < 2
O 4 < (n + 2)!
O 4 < 4
O 2^n < (n + 2)!
O 4 < 6
Is P(2) true?
O Yes
O No
(b) What is P(k)?
O 2^k < k!
O 2^k < (k + 1)!
O k^2 < k!
O k^2 < (k + 1)!
O 2^n < (n + k)!
(c) What is P(k + 1)?
O 2^k + 1 < (k + 2)!
O (k + 1)^2 < (k + 1)!
O 2^n + 1 < (n + k + 1)!
O 2^k + 1 < (k + 1)!
O (k + 1)^2 < (k + 2)!
(d) In a proof by mathematical induction that this inequality holds for every integer n ? 2, what must be shown in the inductive step?
O 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.
O 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.
O 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.
O 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.

Added by Patrick A.

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