+128474 2^200 < n^100 < ( 130n)^50
Note that we can write
2^200 = (2^2)^100 = 4^100 = (4^2)^50 = (16)^50
And we can write
n^100 = (n^2) ^50
So we have that
(16)^50 < (n^2)^50 < (130n)^50 which implies that
16 < n^2 so n > 4 and since we require an integer for n, the solution to this part is [ 5, inf)
And
n^2 < 130n
n^2 - 130n < 0
n(n - 130)< 0
This will be true on the interval ( 0, 130) so....the interval for this part is [ 1, 129 ]
So.....the most restrictive interval that solves this is
[5, 129]
So the number of positive integers is just 129 - 5 + 1 = 125 positive integers
