What is the value of [tex]S_4[/tex] for [tex]\sum_{n=1}^{\infty} 4\left(\frac{1}{4}\right)^{n-1}[/tex] ?
A. [tex]5 \frac{5}{64}[/tex]
B. [tex]5 \frac{1}{4}[/tex]
C. [tex]5 \frac{5}{16}[/tex]
D. [tex]5 \frac{5}{6}[/tex]
Answer (2)
Identify the first term a = 4 and common ratio r = 4 1 of the geometric series.
Apply the formula for the sum of the first 4 terms: S 4 = 1 − 4 1 4 ( 1 − ( 4 1 ) 4 ) .
Simplify the expression: S 4 = 4 3 4 ( 1 − 256 1 ) = 4 3 64 255 .
Calculate the final value: S 4 = 16 85 = 5 16 5 .
Explanation
Understanding the Problem We are given an infinite geometric series and asked to find the sum of its first 4 terms, denoted as S 4 . The series is given by ∑ n = 1 ∞ 4 ( 4 1 ) n − 1 .
Identifying First Term and Common Ratio First, we need to identify the first term ( a ) and the common ratio ( r ) of the geometric series. The general form of a geometric series is a + a r + a r 2 + a r 3 + … . In our case, the series starts with n = 1 , so the first term is a = 4 ( 4 1 ) 1 − 1 = 4 ( 4 1 ) 0 = 4 ( 1 ) = 4 . The common ratio is the factor by which each term is multiplied to get the next term, which is r = 4 1 .
Applying the Sum Formula Now that we have the first term a = 4 and the common ratio r = 4 1 , we can find the sum of the first 4 terms using the formula for the sum of a finite geometric series: S n = 1 − r a ( 1 − r n ) . In our case, we want to find S 4 , so n = 4 . Plugging in the values, we get: S 4 = 1 − 4 1 4 ( 1 − ( 4 1 ) 4 ) .
Simplifying the Expression Let's simplify the expression. First, we calculate ( 4 1 ) 4 = 4 4 1 = 256 1 . Then, we have S 4 = 1 − 4 1 4 ( 1 − 256 1 ) = 4 4 − 1 4 ( 256 256 − 1 ) = 4 3 4 ( 256 255 ) .
Further Simplification Now, we simplify further: S 4 = 4 3 4 ( 256 255 ) = 4 3 64 255 = 64 255 ⋅ 3 4 = 16 255 ⋅ 3 1 = 16 85 .
Converting to Mixed Number Finally, we convert the improper fraction 16 85 to a mixed number. We divide 85 by 16: 85 ÷ 16 = 5 with a remainder of 5. So, 16 85 = 5 16 5 .
Final Answer Therefore, the value of S 4 for the given series is 5 16 5 .
Examples
Understanding geometric series can help in calculating compound interest. For example, if you deposit money into an account that earns interest compounded annually, the amount you have each year forms a geometric sequence. The sum of these amounts over several years can be calculated using the formula for the sum of a geometric series. This is useful for financial planning and understanding the growth of investments. For instance, if you invest $1000 each year with a 5% annual return, the geometric series helps determine the total value of your investment after a certain number of years.
The value of S 4 for the geometric series is 5 16 5 . This is calculated using the sum formula for a geometric series. Therefore, the correct answer option is C.
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