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Find the value of the following expression and round to the nearest integer: sum_(n=1)^(86)20(1.02)^(n) Answer:

Find the value of the following expression and round to the nearest integer:\newline∑n=18620(1.02)n \sum_{n=1}^{86} 20(1.02)^{n} \newlineAnswer:

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Evaluate definite integrals using the chain ruleright chevron icon

Full solution

Q. Find the value of the following expression and round to the nearest integer:\newline∑n=18620(1.02)n \sum_{n=1}^{86} 20(1.02)^{n} \newlineAnswer:
  1. Recognize as geometric series: Recognize the sum as a geometric series.\newlineThe general form of a geometric series is ∑n=0Narn\sum_{n=0}^{N} ar^n, where aa is the first term, rr is the common ratio, and NN is the number of terms.\newlineIn this case, a=20(1.02)a = 20(1.02), r=1.02r = 1.02, and N=86N = 86.
  2. Use formula for sum: Use the formula for the sum of a finite geometric series.\newlineThe sum SS of the first NN terms of a geometric series is given by S=a(1−rN)/(1−r)S = a(1 - r^N) / (1 - r), provided that |r| < 1.\newlineHere, we need to adjust the formula because our series starts at n=1n=1, so we subtract the first term (n=0n=0) from the sum.
  3. Calculate sum using formula: Calculate the sum using the adjusted formula.\newlineS=20(1.02)(1−(1.02)86)/(1−1.02)−20(1.02)0S = 20(1.02)(1 - (1.02)^{86}) / (1 - 1.02) - 20(1.02)^0\newlineSince (1.02)0=1(1.02)^0 = 1, we subtract 2020 from the sum.\newlineS=20(1.02)(1−(1.02)86)/(−0.02)−20S = 20(1.02)(1 - (1.02)^{86}) / (-0.02) - 20
  4. Perform calculations: Perform the calculations.\newlineS=20(1.02)(1−(1.02)86)/(−0.02)−20S = 20(1.02)(1 - (1.02)^{86}) / (-0.02) - 20\newlineS=−20(1.02)(1−(1.02)86)/0.02−20S = -20(1.02)(1 - (1.02)^{86}) / 0.02 - 20\newlineS=−20(1.02)(1−(1.02)86)×50−20S = -20(1.02)(1 - (1.02)^{86}) \times 50 - 20\newlineS=−20×50×(1.02−(1.02)87)−20S = -20 \times 50 \times (1.02 - (1.02)^{87}) - 20\newlineS=−1000×(1.02−(1.02)87)−20S = -1000 \times (1.02 - (1.02)^{87}) - 20
  5. Calculate power of 1.02871.02^{87}: Calculate the power of 1.02871.02^{87}. This step involves using a calculator or a computer to find the value of (1.02)87(1.02)^{87}.
  6. Substitute value of 1.02871.02^{87}: Substitute the value of (1.02)87(1.02)^{87} into the sum.\newlineAssuming the value of (1.02)87(1.02)^{87} is calculated correctly, substitute it back into the expression for SS.\newlineS=−1000×(1.02−(1.02)87)−20S = -1000 \times (1.02 - (1.02)^{87}) - 20
  7. Complete calculation and round: Complete the calculation and round to the nearest integer.\newlineAfter finding the exact value of SS, round it to the nearest integer to get the final answer.

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