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Find the value of the following expression and round to the nearest integer: sum_(n=0)^(31)40(1.02)^(n) Answer:

Find the value of the following expression and round to the nearest integer:\newline∑n=03140(1.02)n \sum_{n=0}^{31} 40(1.02)^{n} \newlineAnswer:

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Q. Find the value of the following expression and round to the nearest integer:\newline∑n=03140(1.02)n \sum_{n=0}^{31} 40(1.02)^{n} \newlineAnswer:
  1. Given series information: We are given a geometric series with the first term a=40a = 40 and common ratio r=1.02r = 1.02. The sum of a finite geometric series is given by the formula Sn=a(1−rn)(1−r)S_n = \frac{a(1 - r^n)}{(1 - r)}, where nn is the number of terms. In this case, we want to find the sum from n=0n=0 to n=31n=31, which means there are 3232 terms.
  2. Calculate sum formula: First, we calculate the sum using the formula for the sum of a geometric series. We have a=40a = 40, r=1.02r = 1.02, and n=32n = 32 terms.S32=40(1−1.0232)(1−1.02)S_{32} = \frac{40(1 - 1.02^{32})}{(1 - 1.02)}
  3. Calculate 1.02321.02^{32}: Now we calculate 1.02321.02^{32} using a calculator.\newline1.0232≈2.03988731.02^{32} \approx 2.0398873
  4. Substitute into formula: Substitute the value of 1.02321.02^{32} into the sum formula.\newlineS32=40(1−2.0398873)/(1−1.02)S_{32} = 40(1 - 2.0398873) / (1 - 1.02)
  5. Calculate numerator and denominator: Calculate the numerator and denominator separately.\newlineNumerator: 40(1−2.0398873)≈40(−1.0398873)≈−41.5954940(1 - 2.0398873) \approx 40(-1.0398873) \approx -41.59549\newlineDenominator: 1−1.02=−0.021 - 1.02 = -0.02
  6. Divide numerator by denominator: Now we divide the numerator by the denominator to find the sum.\newlineS32=−41.59549−0.02≈2079.7745S_{32} = \frac{-41.59549}{-0.02} \approx 2079.7745
  7. Round to nearest integer: Finally, we round the sum to the nearest integer. S32≈2080S_{32} \approx 2080

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