Please show the base case the inductive hypothesis and the i

Please show the base case, the inductive hypothesis, and the inductive step in your solution.

Solution

we need to prove 5^w<w!

Let check it for the base case, w=12
5^12 = 12!
which is true

Now lets assume that,
5^k < k!

Then we need to prove that,
5^(k+1) < (k+1)!
5^k * 5 < (k+1)*k!
since, 5^k < (k+1) , we need to just prove that
5<(k+1)
which is true for k>12

Hence by induction,
5^w<w! is true