Find the sum of the geometric series, if possible.
-$$\sum _ { n = 1 } ^ { 7 } 6 \left( \frac { 2 } { 3 } \right) ^ { n - 1 }$$
A) 35
B) 18
C) $\frac { 4118 } { 243 }$
D) 28

Question

Quizplus · Accepted Answer

The sum of a geometric series can be found using the formula $S_n = \frac{a(1 - r^n)}{1 - r}$, where $S_n$ is the sum of the first $n$ terms, $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. Here, $a = 6$, $r = \frac{2}{3}$, and $n = 7$. Plugging these values into the formula gives $S_7 = \frac{6(1 - (\frac{2}{3})^7)}{1 - \frac{2}{3}} = \frac{6(1 - \frac{128}{2187})}{\frac{1}{3}} = \frac{6 \cdot \frac{2059}{2187}}{\frac{1}{3}} = \frac{4118}{243}$.

Find the Sum of the Geometric Series, If Possible ∑n=176(23)n−1\sum _ { n = 1 } ^ { 7 } 6 \left( \frac { 2 } { 3 } \right) ^ { n - 1 }n=1∑7​6(32​)n−1

Find the Sum of the Geometric Series, If Possible $\sum _ { n = 1 } ^ { 7 } 6 \left( \frac { 2 } { 3 } \right) ^ { n - 1 }$