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sum_(k=0)^(29)-3(0.9)^(k)~~ Choose 1 answer: (A) -4.63*10^(13) (B) -28.7 (c) -1.65 (D) -0.14

∑k=029−3(0.9)k≈ \sum_{k=0}^{29}-3(0.9)^{k} \approx \newlineChoose 11 answer:\newline(A) −4.63⋅1013 -4.63 \cdot 10^{13} \newline(B) −28-28.77\newline(C) −1-1.6565\newline(D) −0-0.1414

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Full solution

Q. ∑k=029−3(0.9)k≈ \sum_{k=0}^{29}-3(0.9)^{k} \approx \newlineChoose 11 answer:\newline(A) −4.63⋅1013 -4.63 \cdot 10^{13} \newline(B) −28-28.77\newline(C) −1-1.6565\newline(D) −0-0.1414
  1. Identify series type: Identify the type of series. The series given is a geometric series because each term is obtained by multiplying the previous term by a common ratio, which is 0.90.9 in this case.
  2. Use formula for sum: Use the formula for the sum of a finite geometric series.\newlineThe sum SS of a geometric series with first term aa, common ratio rr (where |r| < 1), and nn terms is given by the formula:\newlineS=a(1−rn)1−rS = \frac{a(1 - r^n)}{1 - r}\newlineHere, a=−3a = -3 (the first term), r=0.9r = 0.9 (the common ratio), and n=30n = 30 (since we are summing from k=0k=0 to k=29k=29).
  3. Plug values and calculate: Plug the values into the formula and calculate the sum.\newlineS=−3(1−0.930)/(1−0.9)S = -3(1 - 0.9^{30}) / (1 - 0.9)\newlineNow we need to calculate 0.9300.9^{30} and then the rest of the expression.
  4. Calculate 0.9300.9^{30}: Calculate 0.9300.9^{30}. 0.9300.9^{30} is a very small number, and for practical purposes, we can consider it to be close to 00 when subtracted from 11, given the context of the other answer choices. However, for accuracy, we should still calculate it.
  5. Calculate sum using approximation: Calculate the sum using the approximation that 0.9300.9^{30} is close to 00. \newlineS≈−3(1−0)/(1−0.9)S \approx -3(1 - 0) / (1 - 0.9)\newlineS≈−3/0.1S \approx -3 / 0.1\newlineS≈−30S \approx -30
  6. Check answer choices: Check the answer choices for the closest value to our calculation.\newlineThe closest value to −30-30 is (B) −28.7-28.7. Therefore, the sum of the series is approximately −28.7-28.7.

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