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Find the value of the following expression and round to the nearest integer: sum_(n=0)^(24)200(1.22)^(n) Answer:

Find the value of the following expression and round to the nearest integer:\newline∑n=024200(1.22)n \sum_{n=0}^{24} 200(1.22)^{n} \newlineAnswer:

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Full solution

Q. Find the value of the following expression and round to the nearest integer:\newline∑n=024200(1.22)n \sum_{n=0}^{24} 200(1.22)^{n} \newlineAnswer:
  1. Given series information: We are given a geometric series with the first term a=200a = 200 and the common ratio r=1.22r = 1.22. The sum of a finite geometric series can be found using the formula S=a(1−rn)(1−r)S = \frac{a(1 - r^n)}{(1 - r)}, where nn is the number of terms. In this case, n=25n = 25 because we start counting from 00.
  2. Calculate rnr^n: First, we calculate rnr^n, which is 1.22251.22^{25}. This requires a calculator.\newline1.2225≈72.89051.22^{25} \approx 72.8905
  3. Substitute values into formula: Next, we substitute the values into the sum formula for a geometric series:\newlineS=200(1−1.2225)(1−1.22)S = \frac{200(1 - 1.22^{25})}{(1 - 1.22)}
  4. Calculate numerator: Now we calculate the numerator of the fraction: 1−1.2225≈1−72.8905≈−71.89051 - 1.22^{25} \approx 1 - 72.8905 \approx -71.8905
  5. Calculate denominator: Then we calculate the denominator of the fraction: 1−1.22≈−0.221 - 1.22 \approx -0.22
  6. Divide numerator by denominator: Now we divide the numerator by the denominator to find the sum SS:S≈200(−71.8905)(−0.22)S \approx \frac{200(-71.8905)}{(-0.22)}
  7. Perform division: Perform the division to find the sum: S≈200×326.775≈65355S \approx 200 \times 326.775 \approx 65355
  8. Round to nearest integer: Finally, we round the sum to the nearest integer: 6535565355 rounds to 6535565355 since it is already an integer.

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