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gsenviro eNotes educator| Certified Educator

We are given, `n!>` `2^n`  , n `>=` 4

The mathematical induction works by 2 steps, the first step is to show that the given formula is true for base case, i.e. n=1 (here, it would be n=4, since n>=4). 

This step has been done already.

The inductive step involves, assuming that for n=k, the given formula is true, i.e. `k!>2^k`

and then showing that the formula holds for n = k+1.

The first part of the inductive step has already been done, so the Next logical step is showing that the formula holds for n = k+1

For this, we know that (k+1)! = k! (k+1).

Let us multiple both sides of assumed fact (for n=k) by (k+1),

k! (k+1) > `2^k` (k+1).  which is option B.

we can solve for the rest of the question, by using the fact that k+1 > 2 , since k>=4. using this knowledge, we can show that formula holds for k+1.

Thus the next best step is option B.

Hope this helps.

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