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Find the value of the following expression and round to the nearest integer: sum_(n=0)^(25)100(1.24)^(n+1) Answer:

Find the value of the following expression and round to the nearest integer:\newline∑n=025100(1.24)n+1 \sum_{n=0}^{25} 100(1.24)^{n+1} \newlineAnswer:

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Q. Find the value of the following expression and round to the nearest integer:\newline∑n=025100(1.24)n+1 \sum_{n=0}^{25} 100(1.24)^{n+1} \newlineAnswer:
  1. Given Geometric Series: We are given a geometric series with the first term a=100×1.24a = 100 \times 1.24 and the common ratio r=1.24r = 1.24. The sum of the first kk terms of a geometric series is given by the formula Sk=a×(1−rk)/(1−r)S_k = a \times (1 - r^k) / (1 - r) when rr is not equal to 11. Here, we need to find the sum of the series from n=0n = 0 to n=25n = 25, which means we will be summing 2626 terms.
  2. Calculate First Term: First, calculate the first term of the series a=100×1.24a = 100 \times 1.24.
  3. Apply Sum Formula: The first term aa is 124124.
  4. Substitute Values: Now, apply the formula for the sum of the first kk terms of a geometric series: Sk=a⋅(1−rk)/(1−r)S_k = a \cdot (1 - r^k) / (1 - r). Here, k=26k = 26, a=124a = 124, and r=1.24r = 1.24.
  5. Calculate Exponent: Substitute the values into the formula to get S26=124×(1−1.2426)/(1−1.24)S_{26} = 124 \times (1 - 1.24^{26}) / (1 - 1.24).
  6. Substitute Exponent: Calculate 1.24261.24^{26} using a calculator to avoid any potential math errors.
  7. Simplify Expression: The value of 1.24261.24^{26} is approximately 99.53299.532. This is a large number, so it's important to use a calculator to ensure accuracy.
  8. Calculate Sum: Now, substitute 99.53299.532 back into the formula to get S26=124×(1−99.532)/(1−1.24)S_{26} = 124 \times (1 - 99.532) / (1 - 1.24).
  9. Round to Nearest Integer: Simplify the expression to find the sum S26=124×(−98.532)/(−0.24)S_{26} = 124 \times (-98.532) / (-0.24).
  10. Round to Nearest Integer: Simplify the expression to find the sum S26=124×(−98.532)/(−0.24)S_{26} = 124 \times (-98.532) / (-0.24).Calculate the sum S26=124×410.55S_{26} = 124 \times 410.55 (rounded to two decimal places for intermediate steps).
  11. Round to Nearest Integer: Simplify the expression to find the sum S26=124×(−98.532)/(−0.24)S_{26} = 124 \times (-98.532) / (-0.24).Calculate the sum S26=124×410.55S_{26} = 124 \times 410.55 (rounded to two decimal places for intermediate steps).The sum S26S_{26} is approximately 50908.250908.2.
  12. Round to Nearest Integer: Simplify the expression to find the sum S26=124×(−98.532)/(−0.24)S_{26} = 124 \times (-98.532) / (-0.24).Calculate the sum S26=124×410.55S_{26} = 124 \times 410.55 (rounded to two decimal places for intermediate steps).The sum S26S_{26} is approximately 50908.250908.2.Round 50908.250908.2 to the nearest integer to get the final answer.
  13. Round to Nearest Integer: Simplify the expression to find the sum S26=124×(−98.532)/(−0.24)S_{26} = 124 \times (-98.532) / (-0.24).Calculate the sum S26=124×410.55S_{26} = 124 \times 410.55 (rounded to two decimal places for intermediate steps).The sum S26S_{26} is approximately 50908.250908.2.Round 50908.250908.2 to the nearest integer to get the final answer.The rounded value is 5090850908.

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