The sum of 100 first members of a geometric progression is S(100)=1, S(200)=5. Find S(250)

The sum of first 100 members of a geometric progression is S(100)=1,
The sum of first 200 members of geometric progression is S(200)=5.
Find the sum of first 250 members S(250).

Sum of members 101 to 200 (inclusive) = 5 - 1 = 4
=> (sum of members 101 to 200) / (sum of members 1 to 100) = 4
=> member[x+100] / member[x] = 4
=> member[x+50] / member[x] = 2 .... because it's a geometric series
=> (sum of members 51 to 100) / (sum of members 1 to 50) = 2

but (sum of members 51 to 100) + (sum of members 1 to 50) = 1
=> (sum of members 51 to 100) = 2/3
and (sum of members 1 to 50) = 1/3

and since member[x+50] / member[x] = 2 we can now calculate:
(sum of members 101 to 150) = (sum of members 51 to 100) * 2 = 4/3
(sum of members 151 to 200) = (sum of members 101 to 150) * 2 = 8/3
etc