Find the sum of the infinite geometric sequence. $\sum _ { n = 1 } ^ { \infty } 8 - \left( \frac { 1 } { 6 } \right) ^ { n - 1 }$
A) s = $\frac { 5 } { 34 }$
B) s = $\frac { 34 } { 5 }$
C) s = $\frac { 42 } { 5 }$
D) s = $\frac { 46 } { 5 }$
E) s = $\frac { 1 } { 5 }$

Question

Find the sum of the infinite geometric sequence.   $\sum _ { n = 1 } ^ { \infty } 8 - \left( \frac { 1 } { 6 } \right) ^ { n - 1 }$
A) s = $\frac { 5 } { 34 }$
B) s = $\frac { 34 } { 5 }$
C) s = $\frac { 42 } { 5 }$
D) s = $\frac { 46 } { 5 }$
E) s = $\frac { 1 } { 5 }$

Quizplus · Accepted Answer

The given series can be rewritten as $\sum _ { n = 1 } ^ { \infty } 8 - \sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { 6 } \right) ^ { n - 1 }$. The first part is an infinite series of 8s, which does not converge. However, the series is actually $\sum _ { n = 1 } ^ { \infty } \left( 8 - \left( \frac { 1 } { 6 } \right) ^ { n - 1 } \right)$, which is a single geometric series with the first term $a = 8 - \frac{1}{6}^0 = 8 - 1 = 7$ and common ratio $r = -\frac{1}{6}$. The sum of an infinite geometric series is given by $S = \frac{a}{1 - r}$, where $a$ is the first term and $r$ is the common ratio. Here, $a = 7$ and $r = -\frac{1}{6}$, so the sum is $S = \frac{7}{1 - (-\frac{1}{6})} = \frac{7}{1 + \frac{1}{6}} = \frac{7}{\frac{7}{6}} = \frac{7 \times 6}{7} = 6$. The correct interpretation should have been the sum of an infinite geometric series starting with 8 and subtracting fractions of $\frac{1}{6}$ raised to successive powers, which converges to a finite value. The correct sum is found by recognizing the series as a combination of a constant and a geometric series, where the geometric part has a first term of 1 (since $n-1$ starts at 0, making the first term $8 - 1$) and a common ratio of $\frac{1}{6}$. The sum of the geometric part is $\frac{1}{1 - \frac{1}{6}} = \frac{1}{\frac{5}{6}} = \frac{6}{5}$, and adding the constant part (which was incorrectly omitted in the explanation) gives the correct total sum. The mistake was in the interpretation of the series structure and calculation of its sum. The correct approach involves identifying the series as a geometric sequence and applying the formula for the sum of an infinite geometric series correctly, considering both the constant and the variable parts of the series.

Find the Sum of the Infinite Geometric Sequence ∑n=1∞8−(16)n−1\sum _ { n = 1 } ^ { \infty } 8 - \left( \frac { 1 } { 6 } \right) ^ { n - 1 }∑n=1∞​8−(61​)n−1

Find the Sum of the Infinite Geometric Sequence $\sum _ { n = 1 } ^ { \infty } 8 - \left( \frac { 1 } { 6 } \right) ^ { n - 1 }$