Final answer:
The sum of the first eight terms of the given geometric sequence is 255, calculated by using the given sum of the first four and two terms to determine the first term and the common ratio.
Step-by-step explanation:
To find the sum of the first eight terms of the geometric series described, we need to use the given information to determine the first term and the common ratio. We know the sum of the first four terms (S4) is 40 and the sum of the first two terms (S2) is 4.
Using the formula for the sum of the first n terms of a geometric sequence, S=a(1-r^n)/(1-r), where a is the first term and r is the common ratio, we can set up two equations:
- S2 = a(1-r^2)/(1-r) = 4
- S4 = a(1-r^4)/(1-r) = 40
Dividing the second equation by the first equation eliminates a and gives us r^2:
(1-r^4)/(1-r^2) = 40/4 = 10,
From which we solve for r and then find a using either S2 or S4. Once we have a and r, we can find the sum of the first eight terms (S8) using the sum formula:
S8 = a(1-r^8)/(1-r).
For a positive common ratio and satisfying our conditions, we would get r=2 and a=1. Then, applying the values to the sum formula:
S8 = 1(1-2^8)/(1-2) = 255.
Therefore, the sum of the first eight terms of the geometric sequence is 255.