Find the sum of the series.
-$\sum _ { n = 0 } ^ { \infty } \left( \frac { 5 } { 9 ^ { n } } + \frac { 9 } { 2 ^ { n } } \right)$
A) $\frac { 21 } { 2 }$
B) $- \frac { 99 } { 8 }$
C) $\frac { 189 } { 8 }$
D) $- \frac { 3 } { 2 }$

Question

Quizplus · Accepted Answer

The series can be split into two separate geometric series: $\sum _ { n = 0 } ^ { \infty } \frac { 5 } { 9 ^ { n } }$ and $\sum _ { n = 0 } ^ { \infty } \frac { 9 } { 2 ^ { n } }$. The first series has a sum of $\frac{5}{1-\frac{1}{9}} = \frac{45}{8}$ and the second series has a sum of $\frac{9}{1-\frac{1}{2}} = 18$. Adding these gives $\frac{45}{8} + 18 = \frac{189}{8}$.

Find the Sum of the Series ∑n=0∞(59n+92n)\sum _ { n = 0 } ^ { \infty } \left( \frac { 5 } { 9 ^ { n } } + \frac { 9 } { 2 ^ { n } } \right)∑n=0∞​(9n5​+2n9​)

Find the Sum of the Series $\sum _ { n = 0 } ^ { \infty } \left( \frac { 5 } { 9 ^ { n } } + \frac { 9 } { 2 ^ { n } } \right)$