Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges.
-$$7 + \frac { 7 } { 4 } + \frac { 7 } { 16 } + \ldots + \frac { 7 } { 4 ^ { n - 1 } } + \ldots$$
A) $\frac { \left( 1 - \frac { 1 } { 4 ^ { n } } \right) } { 1 - \frac { 1 } { 4 } } ; \frac { 28 } { 5 }$
B) $\frac { 7 \left( 1 - \frac { 1 } { 4 ^ { n - 1 } } \right) } { 1 - \frac { 1 } { 4 } } ; \frac { 28 } { 5 }$
C) $\frac { 7 \left( 1 - \frac { 1 } { 4 ^ { n - 1 } } \right) } { 1 - \frac { 1 } { 4 } } ; \frac { 28 } { 3 }$
D) $\frac { 7 \left( 1 - \frac { 1 } { 4 ^ { n } } \right) } { 1 - \frac { 1 } { 4 } } ; \frac { 28 } { 3 }$

Question

Quizplus · Accepted Answer

The series is a geometric series with first term 7 and common ratio 1/4. The formula for the nth partial sum of a geometric series is $ S_n = a \frac{1 - r^n}{1 - r} $. Applying this, the nth partial sum is $ \frac{7(1 - (1/4)^n)}{1 - 1/4} $. As n approaches infinity, the series converges to $ \frac{7}{3} \times 4 = \frac{28}{3} $.

Find a Formula for the Nth Partial Sum of the Series