Use the formula for S n to find the sum of the first five terms of the geometric sequence.
-$$\frac { 2 } { 3 } , \frac { 8 } { 3 } , \frac { 32 } { 3 } , \frac { 128 } { 3 } , \ldots$$
A) $\frac { 226 } { 5 }$
B) $\frac { 682 } { 15 }$
C) $\frac { 682 } { 3 }$
D) 226

Question

Quizplus · Accepted Answer

The given sequence is a geometric sequence with the first term $a = \frac{2}{3}$ and common ratio $r = 4$. The sum of the first $n$ terms of a geometric sequence is given by $S_n = \frac{a(1 - r^n)}{1 - r}$ for $r \neq 1$. Substituting $a = \frac{2}{3}$, $r = 4$, and $n = 5$, we get $S_5 = \frac{\frac{2}{3}(1 - 4^5)}{1 - 4} = \frac{\frac{2}{3}(1 - 1024)}{-3} = \frac{2(1023)}{9} = \frac{682}{3}$.

Use the Formula for Sn to Find the Sum of the First

Use the Formula for S_n to Find the Sum of the First