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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? Round your final answer to the nearest meter.\newlinem

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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? Round your final answer to the nearest meter.\newlinem
  1. Identify Distances Sequence: Identify the sequence of the distances the monkey travels with each swing. The monkey travels through an arc of 1010 meters 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 1010 meters, and the common ratio (r)(r) is (910)(\frac{9}{10}).
  2. Write Sum Formula: Write the formula for the sum of the first nn terms of a geometric sequence.\newlineThe sum of the first nn terms of a geometric sequence is given by the formula Sn=a1×(1−rn)/(1−r)S_n = a_1 \times (1 - r^n) / (1 - r), where SnS_n is the sum of the first nn terms, a1a_1 is the first term, rr is the common ratio, and nn is the number of terms.
  3. Plug Values for 2525 Swings: Plug the values into the formula to find the total distance after 2525 swings.\newlineWe have a1=10a_1 = 10 meters, r=910r = \frac{9}{10}, and n=25n = 25. Now we can calculate the sum of the first 2525 terms.\newlineS25=10×(1−(910)25)/(1−910)S_{25} = 10 \times \left(1 - \left(\frac{9}{10}\right)^{25}\right) / \left(1 - \frac{9}{10}\right)
  4. Calculate Sum with Values: Calculate the sum using the values provided.\newlineS25=10×(1−(910)25)/(1−910)S_{25} = 10 \times (1 - (\frac{9}{10})^{25}) / (1 - \frac{9}{10})\newlineS25=10×(1−(910)25)/(110)S_{25} = 10 \times (1 - (\frac{9}{10})^{25}) / (\frac{1}{10})\newlineS25=10×(1−(910)25)×10S_{25} = 10 \times (1 - (\frac{9}{10})^{25}) \times 10\newlineS25=100×(1−(910)25)S_{25} = 100 \times (1 - (\frac{9}{10})^{25})
  5. Calculate (9/10)25(9/10)^{25}: Calculate (9/10)25(9/10)^{25} to find the exact value to subtract from 11.(9/10)25(9/10)^{25} is a very small number, so we can use a calculator to find its value.(9/10)25≈0.072(9/10)^{25} \approx 0.072
  6. Subtract and Multiply for Total Distance: Subtract (9/10)25(9/10)^{25} from 11 and multiply by 100100 to find the total distance.\newlineS25=100×(1−0.072)S_{25} = 100 \times (1 - 0.072)\newlineS25=100×0.928S_{25} = 100 \times 0.928\newlineS25≈92.8S_{25} \approx 92.8
  7. Round Final Answer: Round the final answer to the nearest meter.\newlineSince 92.892.8 is closer to 9393 than to 9292, we round up.\newlineS25≈93S_{25} \approx 93 meters

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