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sum_(k=0)^(39)8(-(7)/(8))^(k)~~ Choose 1 answer: (A) 0.24 (B) 4.25 (C) 4.29 (D) 64.31

∑k=0398(−78)k≈ \sum_{k=0}^{39} 8\left(-\frac{7}{8}\right)^{k} \approx \newlineChoose 11 answer:\newline(A) 00.2424\newline(B) 44.2525\newline(C) 44.2929\newline(D) 6464.3131

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Q. ∑k=0398(−78)k≈ \sum_{k=0}^{39} 8\left(-\frac{7}{8}\right)^{k} \approx \newlineChoose 11 answer:\newline(A) 00.2424\newline(B) 44.2525\newline(C) 44.2929\newline(D) 6464.3131
  1. Recognize as geometric series: Recognize the series as a geometric series.\newlineA geometric series has the form ∑k=0nark \sum_{k=0}^{n} ar^k , where a a is the first term and r r is the common ratio.\newlineIn this case, a=8 a = 8 and r=−78 r = -\frac{7}{8} .
  2. Use sum formula: Use the formula for the sum of a finite geometric series.\newlineThe sum of the first n+1 n+1 terms of a geometric series is given by Sn=a(1−rn+1)1−r S_n = \frac{a(1-r^{n+1})}{1-r} , provided r≠1 r \neq 1 .
  3. Apply formula to series: Apply the formula to the given series.\newlineHere, n=39 n = 39 , a=8 a = 8 , and r=−78 r = -\frac{7}{8} .\newlineSo, S39=8(1−(−78)40)1−(−78) S_{39} = \frac{8(1-(-\frac{7}{8})^{40})}{1-(-\frac{7}{8})} .
  4. Calculate sum: Calculate the sum using the formula.\newlineS39=8(1−(−78)40)1+78 S_{39} = \frac{8(1-(-\frac{7}{8})^{40})}{1+\frac{7}{8}} \newlineS39=8(1−(−78)40)158 S_{39} = \frac{8(1-(-\frac{7}{8})^{40})}{\frac{15}{8}} \newlineS39=64(1−(−78)40)15 S_{39} = \frac{64(1-(-\frac{7}{8})^{40})}{15}
  5. Simplify expression: Simplify the expression.\newlineSince (−78)40 (-\frac{7}{8})^{40} is a large negative power of a fraction less than 11, it will be very close to 00.\newlineThus, S39≈64(1−0)15 S_{39} \approx \frac{64(1-0)}{15} \newlineS39≈6415 S_{39} \approx \frac{64}{15}
  6. Perform division: Perform the division to find the approximate sum.\newlineS39≈6415≈4.267 S_{39} \approx \frac{64}{15} \approx 4.267
  7. Round to two decimal places: Round the result to two decimal places as the answers are given in two decimal places.\newlineS39≈4.27 S_{39} \approx 4.27

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