Question

For each positive integer n, let P(n) be the following inequality. 2^n < (n + 1)! (a) What is P(2)? 4 < 2 4 < 6 4 < 4 4 < (n + 2)! 2^n < (n + 2)! Is P(2) true? Yes No (b) What is P(k)? 2^k < (k + 1)! 2^n < (n + k)! k^2 < k! k^2 < (k + 1)! 2^k < k! (c) What is P(k + 1)? (k + 1)^2 < (k + 1)! 2^n + 1 < (n + k + 1)! (k + 1)^2 < (k + 2)! 2^{k+1} < (k + 2)! 2^{k+1} < (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.

          For each positive integer n, let P(n) be the following inequality.
2^n < (n + 1)!
(a) What is P(2)?
4 < 2
4 < 6
4 < 4
4 < (n + 2)!
2^n < (n + 2)!
Is P(2) true?
Yes
No
(b) What is P(k)?
2^k < (k + 1)!
2^n < (n + k)!
k^2 < k!
k^2 < (k + 1)!
2^k < k!
(c) What is P(k + 1)?
(k + 1)^2 < (k + 1)!
2^n + 1 < (n + k + 1)!
(k + 1)^2 < (k + 2)!
2^{k+1} < (k + 2)!
2^{k+1} < (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.

Show more…

For each positive integer n, let P(n) be the following inequality.
2^n < (n + 1)!
(a) What is P(2)?
4 < 2
4 < 6
4 < 4
4 < (n + 2)!
2^n < (n + 2)!
Is P(2) true?
Yes
No
(b) What is P(k)?
2^k < (k + 1)!
2^n < (n + k)!
k^2 < k!
k^2 < (k + 1)!
2^k < k!
(c) What is P(k + 1)?
(k + 1)^2 < (k + 1)!
2^n + 1 < (n + k + 1)!
(k + 1)^2 < (k + 2)!
2^k+1 < (k + 2)!
2^k+1 < (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.

Added by Miriam M.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Julie Silva

07/17/2023

Step 1

(a) What is P(2)? P(2) is the inequality when n = 2: 2(2) < (2 + 1)! 4 < 3! 4 < 6 Since 4 is less than 6, P(2) is true. Show more…

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Thanks for your feedback!

For each positive integer n, let P(n) be the following inequality. 2^n < (n + 1)! (a) What is P(2)? 4 < 2 4 < 6 4 < 4 4 < (n + 2)! 2^n < (n + 2)! Is P(2) true? Yes No (b) What is P(k)? 2^k < (k + 1)! 2^n < (n + k)! k^2 < k! k^2 < (k + 1)! 2^k < k! (c) What is P(k + 1)? (k + 1)^2 < (k + 1)! 2^n + 1 < (n + k + 1)! (k + 1)^2 < (k + 2)! 2^k + 1 < (k + 2)! 2^k + 1 < (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.

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Transcript

00:01 So in this problem, we're told that for each positive integer n, that p of n is going to be represented by the following inequality.

00:07 2 to the n is less than the quantity of n plus 1 factorial.

00:12 So let's start with part a.

00:14 They want us to find p of 2.

00:15 Well, that means we're going to substitute 2 in place of n in our inequality, meaning we're going to have 2 to the second power less than the quantity of 2 plus 1 factorial.

00:26 And now we just need to simplify.

00:27 Well, on the left -hand side, we have 2 squared.

00:29 That'd be 2 times 2, which is 4.

00:32 On the right -hand side, first, let's simplify inside our parentheses.

00:35 2 plus 1 is 3, so that leaves us with 3 factorial.

00:39 Remember, 3 factorial means we need to multiply the integers 3 times 2 times 1.

00:44 Well, 3 times 2 is 6, times 1 is still 6.

00:47 So we're left with 4 is less than 6.

00:50 So the answer choice in this case, i'll call it choice b.

00:54 Now, if we take a look at part b, or part, yeah, part b here, or actually, i didn't realize there was a second part there.

01:02 It says, is p of 2 true? well, is 4 less than 6? yes, that is true.

01:07 So it is a true statement.

01:09 Now we can go on to part b...

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