Given the series:
[tex]40-10+2.5-0.625+...[/tex]You need to remember that a Geometric Series has a Common Ratio, which is the factor between the terms.
In this case, the series is geometric, because every term is found by multiplying the previous one by:
[tex]r=\frac{r_2}{r_1}=\frac{-10}{40}=-\frac{1}{4}[/tex]By definition, the sum of the first "n" terms of Geometric Series, can be calculated by using the following formula:
[tex]S_n=\frac{a_1(1-r^n)}{1-r}[/tex]Where:
-The first term is:
[tex]a_1[/tex]- The number of terms is "n".
- And "r" is the Common Ratio. This must be:
[tex]r\ne1[/tex]In this case, you can identify that:
[tex]\begin{gathered} a_1=40 \\ \\ r=-\frac{1}{4} \\ \\ n=6 \end{gathered}[/tex]Therefore, substituting values into the formula and evaluating, you get:
[tex]\begin{gathered} S_6=\frac{40_{}(1-(-\frac{1}{4})^6)}{1-(-\frac{1}{4})} \\ \\ S_6=\frac{40_{}(1-(-\frac{1}{4})^6)}{1+\frac{1}{4}} \end{gathered}[/tex][tex]S_6=\frac{4095}{128}[/tex]Hence, the answer is:
[tex]S_6=\frac{4095}{128}[/tex]