Given :
[tex]-5,\text{ -2, -}\frac{4}{5},\text{ -}\frac{8}{25},\ldots[/tex]The sequence is a geometric progression. Recall that a G.P has a common ratio such that:
[tex]\frac{Third\text{ term}}{Second\text{ term}}\text{ = }\frac{Second\text{ term}}{First\text{ term}}[/tex]Let's go ahead to find the common ratio (r)
[tex]\begin{gathered} \frac{-\frac{4}{5}}{-2}\text{ = }\frac{-\frac{8}{25}}{-\frac{4}{5}}\text{ = }\frac{2}{5} \\ r\text{ = }\frac{2}{5} \end{gathered}[/tex]To find the sum to infinity of geometric progression whose common ratio(r) is less than 1, we use the formula:
[tex]S_{\infty}\text{ = }\frac{a}{1\text{ - r}}[/tex]The first term (a) of the sequence = -5
Hence, the sum to infinity is:
[tex]\begin{gathered} S_{\infty\text{ }}=\text{ }\frac{-5}{1\text{ - }\frac{2}{5}} \\ =\text{ -}\frac{25}{3} \end{gathered}[/tex]Answer = -25/3
