What is the value of $S_4$ for $\sum_{n=1}^{\infty} 4\left(\frac{1}{4}\right)^{n-1}$?
A. $5 \frac{5}{64}$
B. $5 \frac{1}{4}$
C. $5 \frac{5}{16}$
D. $5 \frac{5}{6}$
Answer (1)
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 n terms: S n = 1 − r a ( 1 − r n ) .
Substitute n = 4 into the formula: S 4 = 1 − 4 1 4 ( 1 − ( 4 1 ) 4 ) .
Calculate S 4 and convert to a mixed number: S 4 = 5 16 5 .
Explanation
Understanding the Problem We are given the infinite geometric series ∑ n = 1 ∞ 4 ( 4 1 ) n − 1 . We want to find the value of S 4 , which represents the sum of the first 4 terms of this series.
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 first term is found by plugging in n = 1 into the expression: 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, we use the formula for the sum of the first n terms of a geometric series, which is given by: S n = 1 − r a ( 1 − r n ) In our case, we want to find S 4 , so we substitute a = 4 , r = 4 1 , and n = 4 into the formula: S 4 = 1 − 4 1 4 ( 1 − ( 4 1 ) 4 )
Calculating the Sum Now, let's calculate S 4 : S 4 = 4 3 4 ( 1 − 256 1 ) = 4 3 4 ( 256 256 − 1 ) = 4 3 4 ( 256 255 ) = 4 3 64 255 To simplify further, we divide the fractions: S 4 = 64 255 ÷ 4 3 = 64 255 × 3 4 = 64 × 3 255 × 4 = 16 × 3 255 = 16 85
Converting to Mixed Number Finally, we convert the improper fraction 16 85 to a mixed number. We divide 85 by 16: 85 = 5 × 16 + 5 . So, 16 85 = 5 16 5 . Therefore, S 4 = 5 16 5 .
Final Answer Thus, the value of S 4 for the given geometric series is 5 16 5 .
Examples
Geometric series are useful in many real-world applications. For example, calculating the total amount of money accumulated over several years with a fixed interest rate involves summing a geometric series. Another application is in physics, where the decay of radioactive substances can be modeled using a geometric series. Understanding geometric series helps in predicting long-term outcomes in finance, physics, and other fields where quantities change by a constant ratio over time. For instance, if you invest $1000 each year with a 5% annual return, the total value of your investments can be calculated using the sum of a geometric series.
