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Find the value of the following expression and round to the nearest integer:

sum_(n=0)^(28)200(1.19)^(n+1)
Answer:

Find the value of the following expression and round to the nearest integer:\newlinen=028200(1.19)n+1 \sum_{n=0}^{28} 200(1.19)^{n+1} \newlineAnswer:

Full solution

Q. Find the value of the following expression and round to the nearest integer:\newlinen=028200(1.19)n+1 \sum_{n=0}^{28} 200(1.19)^{n+1} \newlineAnswer:
  1. Given Series Information: We are given a geometric series with the first term a=200×1.19a = 200 \times 1.19 and the common ratio r=1.19r = 1.19. The number of terms nn is 2929 because the series starts from n=0n = 0 and goes up to n=28n = 28. To find the sum of a geometric series, we use the formula Sn=a(1rn)(1r)S_n = \frac{a(1 - r^n)}{(1 - r)} where SnS_n is the sum of the first nn terms.
  2. Calculate First Term: First, calculate the first term aa of the series: a=200×1.19a = 200 \times 1.19.
  3. Calculate Common Ratio: Now, calculate the common ratio rr which is already given as 1.191.19.
  4. Calculate Sum Formula: Next, calculate the sum of the series using the formula Sn=a(1rn)/(1r)S_n = a(1 - r^n) / (1 - r). We substitute a=200×1.19a = 200 \times 1.19, r=1.19r = 1.19, and n=29n = 29 into the formula.
  5. Perform Calculation: Perform the calculation: S29=(200×1.19)(11.1929)11.19.S_{29} = \frac{(200 \times 1.19)(1 - 1.19^{29})}{1 - 1.19}.
  6. Calculate 1.19291.19^{29}: Calculate 1.19291.19^{29} using a calculator to avoid any manual calculation error.
  7. Calculate Numerator: Subtract 1.19291.19^{29} from 11 to get the numerator of the fraction.
  8. Calculate Denominator: Calculate the denominator of the fraction, which is 11.19=0.191 - 1.19 = -0.19.
  9. Divide Numerator by Denominator: Now, divide the numerator by the denominator to get the sum S29S_{29}.
  10. Round to Nearest Integer: After finding the sum S29S_{29}, round the result to the nearest integer as the question prompt asks for the rounded value.
  11. Unable to Compute: Unfortunately, without a calculator, we cannot compute 1.19291.19^{29} and the subsequent operations to get the exact sum and round it. Therefore, we cannot complete the solution here.

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