Final answer:
The sum of the geometric series 1-2+4-8,..., up to n=8 terms is -85.
Step-by-step explanation:
The student has asked to evaluate the geometric series 1-2+4-8,...,n=8. To find the sum of this series, notice that it is alternating, multiplying by -2 at each step. We can write this as:
∑ (-2)^k from k=0 to k=7.
This is a finite geometric series with the first term a1 = 1 and the common ratio r = -2. The sum of the first n terms of a geometric series is given by the formula:
S_n = a1(1 - r^n) / (1 - r).
Plugging the values into the formula, we get:
S_n = 1(1 - (-2)^8) / (1 - (-2)) = 1(1 - 256) / (1 + 2) = -255 / 3 = -85.
Thus, the sum of the series 1-2+4-8,..., up to n=8 terms is -85.