Final answer:
The sum of the next three terms of the geometric progression, given the sums of the first six terms, can be found using the common ratio. The correct option for the sum of the next three terms in this G.P. is C 4016/7.
Step-by-step explanation:
Given that the sum of the first three terms of a geometric progression (G.P.) is 16 and the sum of the next three terms is 128, we need to determine the sum of the following three terms of the G.P.
Let's denote the first term of the G. P. as a and the common ratio as r. The first three terms of the G.P. can be expressed as a, ar, ar^2 and their sum is given by a + ar + ar^2 = 16. Similarly, the next three terms are ar^3, ar^4, ar^5 and their sum is ar^3 + ar^4 + ar^5 = 128. By dividing the second sum by the first, we get r^3. This means:
(ar^3 + ar^4 + ar^5) / (a + ar + ar^2) = 128/16.
Solving the equation above, we see that r^3 = 8, which implies r = 2. The sum of the following three terms will be ar^6 + ar^7 + ar^8.
By using the common ratio r = 2, we can express the sum of these three terms as ar^6(1 + r + r^2), which simplifies to ar^6 * 7. We already know that ar^3 = 64 (since it's the first term of the second sum of three terms), so ar^6 = (ar^3)^2 = 4096.
Finally, the sum of the next three terms is 4096 * 7 = 28672. Hence, the correct option is C 4016/7.