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Find the value of the following expression and round to the nearest integer: sum_(n=1)^(62)40(1.03)^(n) Answer:

Find the value of the following expression and round to the nearest integer:\newline∑n=16240(1.03)n \sum_{n=1}^{62} 40(1.03)^{n} \newlineAnswer:

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Q. Find the value of the following expression and round to the nearest integer:\newline∑n=16240(1.03)n \sum_{n=1}^{62} 40(1.03)^{n} \newlineAnswer:
  1. Given Series Information: We are given a geometric series with the first term a=40(1.03)1a = 40(1.03)^1 and the common ratio r=1.03r = 1.03. 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 the sum of 6262 terms.
  2. Calculate First Term: First, let's calculate the first term of the series: a=40(1.03)1=40×1.03a = 40(1.03)^1 = 40 \times 1.03.
  3. Calculate Common Ratio: Now, we calculate the common ratio rr, which is 1.031.03. Since it is greater than 11, the series is increasing.
  4. Calculate 6262nd Term: Next, we calculate the 62nd62^{\text{nd}} term of the series using the formula for the nthn^{\text{th}} term of a geometric series, which is an=a⋅r(n−1)a_n = a \cdot r^{(n-1)}. So, a62=40⋅1.03(62−1)=40⋅1.0361a_{62} = 40 \cdot 1.03^{(62-1)} = 40 \cdot 1.03^{61}.
  5. Use Sum Formula: Now we can use the sum formula for a geometric series: S62=a(1−r62)/(1−r)S_{62} = a(1 - r^{62}) / (1 - r). Plugging in the values we have S62=40×1.03(1−1.0362)/(1−1.03)S_{62} = 40 \times 1.03(1 - 1.03^{62}) / (1 - 1.03).
  6. Perform Calculations: We perform the calculations for the sum: S62=40×1.03(1−1.0362)/(1−1.03)=40×1.03(1−1.0362)/(−0.03)S_{62} = 40 \times 1.03(1 - 1.03^{62}) / (1 - 1.03) = 40 \times 1.03(1 - 1.03^{62}) / (-0.03).
  7. Calculate Exponentiation: We need to calculate 1.03621.03^{62}. This is a large exponentiation, and it's best to use a calculator for this step to avoid any math errors.
  8. Substitute Value: After calculating 1.03621.03^{62}, we substitute this value back into the sum formula to get the exact value of the sum.
  9. Round to Nearest Integer: Finally, we round the result to the nearest integer as the question prompt asks for the value rounded to the nearest integer.

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