Evaluate:
$\begin{array}{c}
\sum_{n=1}^{10} 8\left(\frac{1}{4}\right)^{n-1} \
S=[?]\end{array}$
Remember: for a geometric series, $S=\frac{a\left(1-r^n\right)}{1-r}$
Answer (2)
Identify the first term a = 8 , common ratio r = 4 1 , and number of terms n = 10 .
Apply the formula for the sum of a geometric series: S = 1 − r a ( 1 − r n ) .
Substitute the values into the formula: S = 1 − 4 1 8 ( 1 − ( 4 1 ) 10 ) .
Calculate the sum: S = 10.666656494140625 . The final answer is 10.666656494140625 .
Explanation
Problem Analysis We are asked to evaluate the sum of a geometric series. The series is given by ∑ n = 1 10 8 ( 4 1 ) n − 1 .
Identifying Parameters The first term of the series is a = 8 ( 4 1 ) 1 − 1 = 8 ( 4 1 ) 0 = 8 . The common ratio is r = 4 1 . The number of terms is n = 10 .
Formula Introduction The formula for the sum of a geometric series is given by: S = 1 − r a ( 1 − r n ) where:
S is the sum of the series,
a is the first term,
r is the common ratio,
n is the number of terms.
Calculation Now, we substitute the values a = 8 , r = 4 1 , and n = 10 into the formula: S = 1 − 4 1 8 ( 1 − ( 4 1 ) 10 ) S = 4 3 8 ( 1 − 4 10 1 ) S = 4 3 8 ( 1 − 1048576 1 ) S = 4 3 8 ( 1048576 1048575 ) S = 8 × 1048576 1048575 × 3 4 S = 3 32 × 1048576 1048575 S = 3145728 33554400 S = 10.666656494140625
Final Answer Therefore, the sum of the geometric series is approximately 10.666656494140625 .
Examples
Geometric series are incredibly useful in finance for calculating the future value of an annuity, where regular payments grow over time with a consistent interest rate. For instance, if you invest a fixed amount every year into a retirement account that earns a steady annual return, the total value of your investment can be calculated using the formula for the sum of a geometric series. This helps in predicting long-term investment growth and planning financial goals.
The sum of the series S = ∑ n = 1 10 8 ( 4 1 ) n − 1 can be calculated using the geometric series formula. The total sum is approximately 10.67 .
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