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Find the value of the following expression and round to the nearest integer: sum_(n=1)^(28)100(1.07)^(n) Answer:

Find the value of the following expression and round to the nearest integer:\newline∑n=128100(1.07)n \sum_{n=1}^{28} 100(1.07)^{n} \newlineAnswer:

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Full solution

Q. Find the value of the following expression and round to the nearest integer:\newline∑n=128100(1.07)n \sum_{n=1}^{28} 100(1.07)^{n} \newlineAnswer:
  1. Given Geometric Series: We are given a geometric series with the first term a=100(1.07)1a = 100(1.07)^1 and the common ratio r=1.07r = 1.07. 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 will use this formula to find the sum of the series from n=1n=1 to n=28n=28.
  2. Calculate First Term: First, we calculate the first term of the series: a=100(1.07)1=100×1.07=107a = 100(1.07)^1 = 100 \times 1.07 = 107.
  3. Calculate Common Ratio: Next, we calculate the common ratio rr, which is given as 1.071.07. Since it is the same for each term, we do not need to perform any additional calculations for rr.
  4. Use Sum Formula: Now, we will use the formula for the sum of the first nn terms of a geometric series to find the sum S28S_{28}. Plugging in the values, we get S28=107(1−1.0728)/(1−1.07)S_{28} = 107(1 - 1.07^{28}) / (1 - 1.07).
  5. Calculate Numerator: We calculate the numerator of the formula: 1−1.07281 - 1.07^{28}. Using a calculator, we find that 1.0728≈7.4351.07^{28} \approx 7.435. Therefore, the numerator becomes 1−7.435=−6.4351 - 7.435 = -6.435.
  6. Calculate Denominator: We calculate the denominator of the formula: 1−1.07=−0.071 - 1.07 = -0.07.
  7. Divide Numerator by Denominator: Now we divide the numerator by the denominator: S28=107×(−6.435)/(−0.07)S_{28} = 107 \times (-6.435) / (-0.07). This simplifies to S28=107×91.93S_{28} = 107 \times 91.93.
  8. Calculate Final Sum: Multiplying 107107 by 91.9391.93, we get S28≈9836.61S_{28} \approx 9836.61.
  9. Round to Nearest Integer: Finally, we round the sum to the nearest integer. The sum S28≈9836.61S_{28} \approx 9836.61 rounds to 98379837.

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