Observe that as the series progresses, the term decreases by 1/4. To show this, observe the first four terms of the series below.
-200 = (1/4)(-800)
-50 = (1/4)(-200)
-12.5 = (1/4)(-50)
Since we have a common ratio, r, of 1/4, we can use the properties of a geometric series to find the 8th term of the series.
Recall that to find the sum of the nth term of a geometric series, we have
[tex] S_{n} = a(\frac{1-r^{n}}{1-r}) [/tex]
where a is the first term of the series and r is the ratio.
So, for the first eight terms, we have
[tex] S_{8} = -800(\frac{1-(\frac{1}{4})^8}{1- \frac{1}{4}}) [/tex]
[tex] S_{8} \approx -1066.65 [/tex]
Therefore, the sum of the 8th series is approximately -1066.65.
Answer: -1066.65
