Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence.
-$$\sum _ { i = 1 } ^ { 3 } \left( \frac { 2 } { 3 } \right) ^ { i + 1 }$$
A) $\frac { 76 } { 81 }$
B) $\frac { 38 } { 27 }$
C) $\frac { 80 } { 81 }$
D) $\frac { 520 } { 81 }$

Question

Quizplus · Accepted Answer

The sum of the first n terms of a geometric sequence is given by $S_n = \frac{a(1 - r^n)}{1 - r}$, where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. For this sequence, $a = \left(\frac{2}{3}\right)^2 = \frac{4}{9}$, $r = \frac{2}{3}$, and $n = 3$. Plugging these values into the formula gives $S_3 = \frac{\frac{4}{9}(1 - (\frac{2}{3})^3)}{1 - \frac{2}{3}} = \frac{\frac{4}{9}(1 - \frac{8}{27})}{\frac{1}{3}} = \frac{\frac{4}{9} \cdot \frac{19}{27}}{\frac{1}{3}} = \frac{76}{81}$.

Find the Indicated Sum ∑i=13(23)i+1\sum _ { i = 1 } ^ { 3 } \left( \frac { 2 } { 3 } \right) ^ { i + 1 }i=1∑3​(32​)i+1

Find the Indicated Sum $\sum _ { i = 1 } ^ { 3 } \left( \frac { 2 } { 3 } \right) ^ { i + 1 }$