Find the sum of the finite geometric sequence.
$\sum _ { n = 1 } ^ { 10 } \left( - \frac { 3 } { 4 } \right) ^ { n - 1 }$
A) $\frac { 1,225,723 } { 262,144 }$
B) $\frac { 141,361 } { 262,144 }$
C) $\frac { 1,068,259 } { 262,144 }$
D) $- \frac { 1,225,723 } { 262,144 }$
E) $- \frac { 1,068,259 } { 262,144 }$

Question

Find the sum of the finite geometric sequence.  
  $\sum _ { n = 1 } ^ { 10 } \left( - \frac { 3 } { 4 } \right) ^ { n - 1 }$  
A)  $\frac { 1,225,723 } { 262,144 }$
B)  $\frac { 141,361 } { 262,144 }$
C)  $\frac { 1,068,259 } { 262,144 }$
D)  $- \frac { 1,225,723 } { 262,144 }$
E)  $- \frac { 1,068,259 } { 262,144 }$

Quizplus · Accepted Answer

The sum of a finite 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, $a = \left(-\frac{3}{4}\right)^0 = 1$, $r = -\frac{3}{4}$, and $n = 10$. Plugging these values into the formula gives $S_{10} = \frac{1(1 - (-\frac{3}{4})^{10})}{1 - (-\frac{3}{4})} = \frac{1 - (-\frac{3}{4})^{10}}{1 + \frac{3}{4}} = \frac{1 - (\frac{3}{4})^{10}}{\frac{7}{4}}$. Simplifying this expression gives the correct answer as $\frac{141,361}{262,144}$.

Find the Sum of the Finite Geometric Sequence ∑n=110(−34)n−1\sum _ { n = 1 } ^ { 10 } \left( - \frac { 3 } { 4 } \right) ^ { n - 1 }∑n=110​(−43​)n−1

Find the Sum of the Finite Geometric Sequence $\sum _ { n = 1 } ^ { 10 } \left( - \frac { 3 } { 4 } \right) ^ { n - 1 }$