Given the series 1,600 – 400 + 100 – 25 + …
The series is a geometric series because it has a common ratio
The common ratio is
[tex]\begin{gathered} r=\frac{-400}{1600} \\ r=\frac{-1}{4} \end{gathered}[/tex][tex]\begin{gathered} \text{ For a geometric series, the sun of nth term is given by the formula} \\ s_n=\frac{a(1-r^n)}{1-r} \\ Where\text{ a = first term = 1600} \\ r=\text{common ratio = -1/4 = -0.25} \\ n=\text{ number of terms} \\ \text{For this question, n=8} \end{gathered}[/tex][tex]\begin{gathered} s_8=\frac{1600(1-(-0.25)^8)}{1-(-0.25)}_{}_{} \\ s_8=\frac{1600(1-0.0000152587)}{1.25}=\frac{1600(0.9999847)}{1.25} \\ s_8=\frac{1599.97558}{1.25} \\ \\ s_8=1279.9804 \end{gathered}[/tex]
To the nearest thousandth, we have 1,279.980
Hence, The 8th partial sum of the series is 1,279.980 to the nearest thousandth
