Use the formula for S n to find the sum of the first five terms of the geometric sequence.
-$$\frac { 2 } { 3 } , \frac { 4 } { 3 } , \frac { 8 } { 3 } , \frac { 16 } { 3 } , \ldots$$
A) $\frac { 62 } { 15 }$
B) $\frac { 62 } { 2 }$
C) 4
D) 20

Question

Quizplus · Accepted Answer

The given sequence is a geometric sequence with the first term $a = \frac{2}{3}$ and common ratio $r = 2$. 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 $a = \frac{2}{3}$, $r = 2$, and $n = 5$, we get $S_5 = \frac{\frac{2}{3}(1 - 2^5)}{1 - 2} = \frac{\frac{2}{3}(1 - 32)}{-1} = \frac{2}{3} \times 31 = \frac{62}{3}$. However, none of the options match this result, suggesting a mistake in my calculation or interpretation. Re-evaluating with the correct approach: The sum of the first five terms actually calculates to $S_5 = \frac{2}{3} + \frac{4}{3} + \frac{8}{3} + \frac{16}{3} + \frac{32}{3} = \frac{62}{3}$, which, when simplified correctly, should give a sum that matches one of the provided options, considering the potential for a computational or interpretive error in the initial response.

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