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Find the value of the following expression and round to the nearest integer: sum_(n=1)^(25)10(0.97)^(n-1) Answer:

Find the value of the following expression and round to the nearest integer:\newline∑n=12510(0.97)n−1 \sum_{n=1}^{25} 10(0.97)^{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=12510(0.97)n−1 \sum_{n=1}^{25} 10(0.97)^{n-1} \newlineAnswer:
  1. Given Series Information: We are given a geometric series with the first term a=10a = 10 and common ratio r=0.97r = 0.97. The sum of the first nn terms of a 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. We need to find S25S_{25}.
  2. Plug in Values: First, let's plug in the values into the formula: S25=10(1−0.9725)/(1−0.97)S_{25} = 10(1 - 0.97^{25}) / (1 - 0.97).
  3. Calculate 0.97250.97^{25}: Now, we calculate the value of 0.97250.97^{25} using a calculator.\newline0.9725≈0.47675225560.97^{25} \approx 0.4767522556
  4. Substitute Value Back: Substitute the value of 0.97250.97^{25} back into the sum formula: S25=10(1−0.4767522556)/(1−0.97)S_{25} = 10(1 - 0.4767522556) / (1 - 0.97).
  5. Perform Subtraction: Perform the subtraction in the numerator and the denominator: S25=10(0.5232477444)0.03S_{25} = \frac{10(0.5232477444)}{0.03}.
  6. Calculate Numerator: Now, we calculate the numerator: 10×0.5232477444≈5.23247744410 \times 0.5232477444 \approx 5.232477444.
  7. Divide by Denominator: Next, we divide by the denominator: S25=5.2324774440.03S_{25} = \frac{5.232477444}{0.03}.
  8. Perform Division: Perform the division to find the sum: S25≈174.4159148S_{25} \approx 174.4159148.
  9. Round to Nearest Integer: Finally, we round the sum to the nearest integer: S25≈174S_{25} \approx 174 (since 0.41591480.4159148 is less than 0.50.5, we round down).

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