1 Answer | Add Yours
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.
We’ve answered 319,655 questions. We can answer yours, too.Ask a question