For the given geometric sequence, find the limit of the infinite series, if it exists.
-$5 , \frac { 5 } { 3 } , \frac { 5 } { 9 } , \frac { 5 } { 27 } , \ldots$
A) 5

B) $\frac { 20 } { 3 }$

C) $\frac { 15 } { 2 }$

D) $\frac { 5 } { 3 }$

Question

For the given geometric sequence, find the limit of the infinite series, if it exists.
-$5 , \frac { 5 } { 3 } , \frac { 5 } { 9 } , \frac { 5 } { 27 } , \ldots$
A) 5

B) $\frac { 20 } { 3 }$

C) $\frac { 15 } { 2 }$

D) $\frac { 5 } { 3 }$

Quizplus · Accepted Answer

Notice that the common ratio between two consecutive terms in the sequence is $\frac {1}{3}$. Let the infinite series be denoted as S, then 
$S = 5 + \frac{5}{3} + \frac{5}{9} + \frac{5}{27} + \cdots$
Multiplying both sides by the common ratio:
$ \frac{1}{3} S = \frac{5}{3} + \frac{5}{9} + \frac{5}{27} + \frac{5}{81} + \cdots$
Subtracting these equations gives:
$ S - \frac{1}{3}S = 5$
Solving for S, we get:
$S = \boxed{\frac{15}{2}}$
Thus, the answer is choice C).

For the Given Geometric Sequence, Find the Limit of the Infinite