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Find the value of the following expression and round to the nearest integer: sum_(n=0)^(93)400(0.94)^(n+1) Answer:

Find the value of the following expression and round to the nearest integer:\newline∑n=093400(0.94)n+1 \sum_{n=0}^{93} 400(0.94)^{n+1} \newlineAnswer:

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Full solution

Q. Find the value of the following expression and round to the nearest integer:\newline∑n=093400(0.94)n+1 \sum_{n=0}^{93} 400(0.94)^{n+1} \newlineAnswer:
  1. Identify Geometric Series: The given expression is a geometric series where the first term a1a_1 is 400×0.94400 \times 0.94, the common ratio rr is 0.940.94, and the number of terms nn is 9494 (since we start counting from 00 up to 9393).
  2. Calculate Sum Formula: To find the sum of a geometric series, we use the formula S=a1×(1−rn)/(1−r)S = a_1 \times (1 - r^n) / (1 - r), where SS is the sum of the series.
  3. Calculate First Term: First, calculate the first term a1a_1: a1=400×0.94=376a_1 = 400 \times 0.94 = 376.
  4. Calculate rnr^n: Next, calculate rnr^n: rn=0.9494r^n = 0.94^{94}. This calculation might be difficult to do exactly without a calculator, but we can estimate it to be very small since 0.940.94 is less than 11 and raised to a high power.
  5. Plug Values into Formula: Now, we can plug the values into the sum formula: S=376×(1−0.9494)/(1−0.94)S = 376 \times (1 - 0.94^{94}) / (1 - 0.94).
  6. Approximate rnr^n: Since 0.94940.94^{94} is very small, we can approximate it to 00 for the purpose of this calculation. Therefore, the formula simplifies to S≈3761−0.94S \approx \frac{376}{1 - 0.94}.
  7. Calculate Denominator: Calculate the denominator: 1−0.94=0.061 - 0.94 = 0.06.
  8. Divide First Term: Now, divide the first term by the denominator: S≈3760.06S \approx \frac{376}{0.06}.
  9. Perform Division: Perform the division: S≈3760.06≈6266.67S \approx \frac{376}{0.06} \approx 6266.67.
  10. Round Final Result: Finally, round the result to the nearest integer: S≈6267S \approx 6267.

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