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A monkey is swinging from a tree. On the first swing, she passes through an arc of 10m. With each swing, she passes through an arc (9)/(10) the length of the previous swing. What is the total distance the monkey has traveled when she completes her 25^("th ") swing? Round your final answer to the nearest meter. ◻ m

A monkey is swinging from a tree. On the first swing, she passes through an arc of 10 m 10 \mathrm{~m} . With each swing, she passes through an arc 910 \frac{9}{10} the length of the previous swing.\newlineWhat is the total distance the monkey has traveled when she completes her 25th  25^{\text {th }} swing?\newlineRound your final answer to the nearest meter.\newline□ \square m \mathrm{m}

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Q. A monkey is swinging from a tree. On the first swing, she passes through an arc of 10 m 10 \mathrm{~m} . With each swing, she passes through an arc 910 \frac{9}{10} the length of the previous swing.\newlineWhat is the total distance the monkey has traveled when she completes her 25th  25^{\text {th }} swing?\newlineRound your final answer to the nearest meter.\newline□ \square m \mathrm{m}
  1. Identify Distances Sequence: Identify the sequence of the distances the monkey travels with each swing. The monkey travels through an arc of 10 m10\,\text{m} on the first swing. Each subsequent swing is (910)(\frac{9}{10}) times the length of the previous swing. This creates a geometric sequence where the first term (a1)(a_1) is 10 m10\,\text{m}, and the common ratio (r)(r) is (910)(\frac{9}{10}).
  2. Calculate Total Distance: Calculate the total distance traveled using the formula for the sum of the first nn terms of a geometric sequence.\newlineThe sum of the first nn terms (SnS_n) of a geometric sequence is given by the formula:\newlineSn=a1×(1−rn)/(1−r)S_n = a_1 \times (1 - r^n) / (1 - r), where nn is the number of terms.\newlineIn this case, a1=10 ma_1 = 10\,m, r=(9/10)r = (9/10), and n=25n = 25.
  3. Substitute Values and Calculate: Substitute the values into the formula and calculate the sum. \newlineSn=10×(1−(910)25)/(1−(910))S_n = 10 \times (1 - (\frac{9}{10})^{25}) / (1 - (\frac{9}{10}))
  4. Calculate (9/10)25(9/10)^{25}: Perform the calculations.\newlineSn=10×(1−(9/10)25)/(1/10)S_n = 10 \times (1 - (9/10)^{25}) / (1/10)\newlineFirst, calculate (9/10)25(9/10)^{25}.
  5. Continue Calculation of SnS_n: Calculate (910)25(\frac{9}{10})^{25} using a calculator to ensure accuracy.\newline(910)25≈0.072(\frac{9}{10})^{25} \approx 0.072
  6. Complete Calculation: Continue with the calculation of SnS_n. \newlineSn=10×(1−0.072)/(1/10)S_n = 10 \times (1 - 0.072) / (1/10)\newlineSn=10×(0.928)/(1/10)S_n = 10 \times (0.928) / (1/10)
  7. Round Final Answer: Complete the calculation.\newlineSn=10×0.928×10S_n = 10 \times 0.928 \times 10\newlineSn=92.8S_n = 92.8
  8. Round Final Answer: Complete the calculation.\newlineSn=10×0.928×10S_n = 10 \times 0.928 \times 10\newlineSn=92.8S_n = 92.8Round the final answer to the nearest meter.\newlineThe total distance traveled by the monkey, rounded to the nearest meter, is approximately 9393 meters.

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