can you check my answer?

2.Which of the following shows the best next step to prove the following by mathematical induction?

n!>2^n, n≥4

1. When n=1 the formula is valid because

4!>2^4
24>16
2.Assuming that k!>2^k
show that (k+1)!>2^k+1
(k+1)!=k!(k+1)

a.k!(k+1)>k!(k+k)
b.k!(k+1)>2^k(k+1)
c.k+1≥5
d.(k+k)>(k+1)!

answer:a

since we're dealing with powers of 2, I'd pick B

Besides, if k>1, (k+1) ≯ (k+k)

To check your answer, let's go through the steps of the mathematical induction proof:

1. Base case:
You correctly state that when n = 1, the formula is valid because 4! > 2^4. This is correct.

2. Inductive hypothesis:
You assume that for some value k, k! > 2^k. This is also correct.

3. Inductive step:
The next step is to prove that (k+1)! > 2^(k+1).

You start by expanding (k+1)! as k!(k+1), which is correct. Now let's analyze the options:

a. k!(k+1) > k!(k+k):
This option is incorrect, as it does not contribute to proving the statement (k+1)! > 2^(k+1).

b. k!(k+1) > 2^k(k+1):
This option is correct. By assuming k! > 2^k, multiplying both sides of the inequality by (k+1) maintains the inequality. Therefore, you have successfully shown that (k+1)! > 2^(k+1).

c. k+1 ≥ 5:
This option is unrelated to the inductive step of proving the statement.

d. (k+k) > (k+1)!:
This option is also unrelated to the inductive step.

Therefore, the correct answer is b, and your initial answer of a was incorrect.

Remember to carefully consider each option and choose the one that supports the inductive step.