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

Elementary and Intermediate Algebra

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

Instant Answer

Solved by Expert Oswaldo Jiménez

09/06/2023

Step 1

(2) We want to prove this inequality by mathematical induction. (3) Base case: We need to check if $P(2)$ is true. $P(2): 2(2) < (2+1)! \Rightarrow 4 < 3! \Rightarrow 4 < 6$, which is true. (4) Inductive step: We need to show that if $P(k)$ is true for some Show more…

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

00:02 For each positive integer n, let pn be the following inequality 2 raised to the nth power less than the factorial of n plus 1.

00:19 In part a we talk about what is p2 and if that statement is true.

00:27 In part b we talk about what is pk.

00:30 In part c what is pk plus 1.

00:33 In part d in a proof by mathematical induction that this inequality holds for every integer n greater than or equal to 2 what must be shown in the inductive step.

00:47 So let's see part a what is p2.

00:52 So remember here the statement in this case an inequality depends on the value of n.

01:00 So we talk about p of n for a positive integer n refers to this inequality.

01:09 In other words the argument let's say like that of the proposition p the argument n will be the exponent of 2 on the left hand side of the inequality and will be the value for which we add 1 and then take the factorial of that sum.

01:33 So to have to find p2 we only replace n by 2 in this inequality right here.

01:40 So we get 2 raised to the second power because n equal 2.

01:43 The argument will be the exponent of 2 so it's 2 raised to the second power or 2 square less than 2 which is n plus 1 factorial of that.

01:55 So we'll get 2 square less than 2 plus 1 factorial that's the inequality we get when n equal 2.

02:13 And that's the same as 2 square is 4 on the left less than and we have 3 factorial on the right and that's equivalent to 4 less than if we calculate 3 factorial is 3 times 2 times 1 is 6.

02:30 So this is p2 4 less than 6 and i'm going to say which is the correct options because you have been given options you have 5 options and the correct options is the third from top to bottom.

02:53 So third option from top to bottom...

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