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Find the value of the following expression and round to the nearest integer: sum_(n=1)^(28)500(0.9)^(n) Answer:

Find the value of the following expression and round to the nearest integer:\newline∑n=128500(0.9)n \sum_{n=1}^{28} 500(0.9)^{n} \newlineAnswer:

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Q. Find the value of the following expression and round to the nearest integer:\newline∑n=128500(0.9)n \sum_{n=1}^{28} 500(0.9)^{n} \newlineAnswer:
  1. Given Geometric Series: We are given the sum of a geometric series with the first term a1=500×0.9 a_1 = 500 \times 0.9 and the common ratio r=0.9 r = 0.9 . The formula for the sum of the first n terms of a geometric series is Sn=a11−rn1−r S_n = a_1 \frac{1 - r^n}{1 - r} . We will use this formula to find the sum of the first 2828 terms.
  2. Calculate First Term: First, we calculate the first term a1=500×0.9=450 a_1 = 500 \times 0.9 = 450 .
  3. Calculate Common Ratio to the Power: Next, we calculate r28=0.928 r^{28} = 0.9^{28} . This requires a calculator or computational tool.
  4. Substitute into Sum Formula: Now we substitute a1 a_1 , r r , and r28 r^{28} into the sum formula: S28=4501−0.9281−0.9 S_{28} = 450 \frac{1 - 0.9^{28}}{1 - 0.9} .
  5. Calculate Denominator: We calculate the denominator 1−0.9=0.1 1 - 0.9 = 0.1 .
  6. Calculate Numerator: We then calculate the numerator 1−0.928 1 - 0.9^{28} using the previously found value for 0.928 0.9^{28} .
  7. Find Sum: After finding the numerator, we divide it by the denominator to find S28 S_{28} .
  8. Round to Nearest Integer: Finally, we round the result to the nearest integer to get our final answer.

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