Question: Which of the following can be used to evaluate this series? \[ \sum_{k=1}^{8} 5…
Which of the following can be used to evaluate this series?
k=1∑85(32)k−1
- 5(1−321−(32)9)
- 5(1−321−(32)7)
- 5(1−321−(32)8)
- 5(1−321−(32)8)
Solution
To evaluate the given series, we recognize it as a geometric series: k=1∑85(32)k−1 The formula for the sum of a geometric series is: Sn=a1−r1−rn where a is the first term and r is the common ratio. 1. Identify the first term a and the common ratio r: The first term when k=1 is: a=5(32)1−1=5 The common ratio r is: r=32 2. Use the formula for the sum of the first n terms, where n=8, a=5, and r=32: S8=51−321−(32)8 Simplifying the expression for S8, we have: S8=5(311−(32)8) This matches the answer choice: 5(1−321−(32)8) Thus, the correct expression to evaluate this series is: 5(1−321−(32)8) Therefore, the correct choice is the fourth option.