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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=185(23)k1 \sum_{k=1}^{8} 5 \left(\frac{2}{3}\right)^{k-1}

  1. 5(1(23)9123)5 \left(\frac{1-\left(\frac{2}{3}\right)^{9}}{1-\frac{2}{3}}\right)
  2. 5(1(23)7123)5 \left(\frac{1-\left(\frac{2}{3}\right)^{7}}{1-\frac{2}{3}}\right)
  3. 5(1(23)8123)5 \left(\frac{1-\left(\frac{2}{3}\right)^{8}}{1-\frac{2}{3}}\right)
  4. 5(1(23)8123)5 \left(\frac{1-\left(\frac{2}{3}\right)^{8}}{1-\frac{2}{3}}\right)

Solution

To evaluate the given series, we recognize it as a geometric series: k=185(23)k1 \sum_{k=1}^{8} 5 \left(\frac{2}{3}\right)^{k-1} The formula for the sum of a geometric series is: Sn=a1rn1r S_n = a \frac{1 - r^n}{1 - r} where a a is the first term and r r is the common ratio. 1. Identify the first term a a and the common ratio r r : The first term when k=1 k = 1 is: a=5(23)11=5 a = 5 \left(\frac{2}{3}\right)^{1-1} = 5 The common ratio r r is: r=23 r = \frac{2}{3} 2. Use the formula for the sum of the first n n terms, where n=8 n = 8 , a=5 a = 5 , and r=23 r = \frac{2}{3} : S8=51(23)8123 S_8 = 5 \frac{1 - \left(\frac{2}{3}\right)^8}{1 - \frac{2}{3}} Simplifying the expression for S8 S_8 , we have: S8=5(1(23)813) S_8 = 5 \left(\frac{1 - \left(\frac{2}{3}\right)^8}{\frac{1}{3}}\right) This matches the answer choice: 5(1(23)8123) 5 \left(\frac{1 - \left(\frac{2}{3}\right)^8}{1 - \frac{2}{3}}\right) Thus, the correct expression to evaluate this series is: 5(1(23)8123) 5 \left(\frac{1 - \left(\frac{2}{3}\right)^8}{1 - \frac{2}{3}}\right) Therefore, the correct choice is the fourth option.

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