Which of the following can be used to evaluate the series [tex]$\sum_{n=1}^5 6(0.6)^{n-1} ?$[/tex]
A. [tex]$0.6\left(\frac{1-(6)^5}{1-0.6}\right)$[/tex]
B. [tex]$0.6\left(\frac{1-(6)^5}{1-6}\right)$[/tex]
C. [tex]$6\left(\frac{1-(0.6)^5}{1-0.6}\right)$[/tex]
D. [tex]$6\left(\frac{1-(0.6)^6}{1-6}\right)$[/tex]
Answer (2)
Identify the first term a = 6 , common ratio r = 0.6 , and number of terms n = 5 .
Apply the formula for the sum of a geometric series: S n = a 1 − r 1 − r n .
Substitute the values into the formula: S 5 = 6 ( 1 − 0.6 1 − ( 0.6 ) 5 ) .
The correct expression is 6 ( 1 − 0.6 1 − ( 0.6 ) 5 ) .
Explanation
Identifying the Series We are asked to identify the correct expression for evaluating the series ∑ n = 1 5 6 ( 0.6 ) n − 1 . This is a geometric series with first term a = 6 ( 0.6 ) 1 − 1 = 6 , common ratio r = 0.6 , and number of terms n = 5 . The formula for the sum of a geometric series is given by S n = a 1 − r 1 − r n .
Applying the Formula Substituting the values a = 6 , r = 0.6 , and n = 5 into the formula, we get: S 5 = 6 ( 1 − 0.6 1 − ( 0.6 ) 5 )
Finding the Correct Option Comparing this result with the given options, we find that the correct expression is 6 ( 1 − 0.6 1 − ( 0.6 ) 5 ) .
Examples
Geometric series are useful in many real-world applications, such as calculating the total amount of money earned over several years with a consistent percentage increase, determining the depreciation of an asset over time, or modeling the spread of a disease. For example, if you invest $1000 each year with a 5% annual return, the geometric series can help you calculate the total value of your investments after a certain number of years. Understanding geometric series helps in making informed financial decisions and predicting future outcomes in various scenarios.
The correct expression to evaluate the series ∑ n = 1 5 6 ( 0.6 ) n − 1 is the option C: 6 ( 1 − 0.6 1 − ( 0.6 ) 5 ) . This formula utilizes the properties of geometric series where the first term is 6, the common ratio is 0.6, and there are 5 terms. Upon substitution, this expression correctly represents the sum of the series.
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