Find the Indicated Partial Sum of the Series $\sum _ { i = 1 } ^ { \infty } 5 \left( \frac { 1 } { 3 } \right) ^ { i }$

Essay

Find the indicated partial sum of the series. $\sum _ { i = 1 } ^ { \infty } 5 \left( \frac { 1 } { 3 } \right) ^ { i }$ fourth partial sum
A) $\frac { 5 } { 81 }$
B) $\frac { 605 } { 81 }$
C) $\frac { 200 } { 81 }$
D) $\frac { 65 } { 81 }$
E) $\frac { 205 } { 81 }$
Use mathematical induction to prove the property for all positive integers $n$ .
$\left[ a ^ { n } \right] ^ { 3 } = a ^ { 3 n }$