Solution:
Given:
[tex]5,-2,\frac{4}{5},\ldots[/tex]
To get the sum of the first 7 terms, the formula below is used;
[tex]S_n=\frac{a_1-a_1r^n}{1-r}[/tex]
where;
[tex]\begin{gathered} n=7 \\ a_1\text{ is the first term = 5} \\ r\text{ is the co}mmon\text{ ratio=}\frac{-2}{5} \end{gathered}[/tex]
Hence,
[tex]\begin{gathered} S_n=\frac{a_1-a_1r^n}{1-r} \\ S_7=\frac{5-5(-\frac{2}{5})^7}{1-(-\frac{2}{5})} \\ S_7=\frac{5-5(-0.4)^7}{1+\frac{2}{5}} \\ S_7=\frac{5-5(-0.0016384)}{1+0.4} \\ S_7=\frac{5+0.008192}{1.4} \\ S_7=\frac{5.008192}{1.4} \\ S_7=3.57728 \end{gathered}[/tex]
Therefore, the sum of the first 7 terms is 3.57728
