A museum contains a large pendulum. Due to air resistance and friction, each swing of the pendulum is a little shorter than the previous one. Suppose the first swing of the pendulum has a length of 25,000 mm and each return swing loses 4% of its length until eventually the pendulum comes to a complete stop. Answer parts A, B, C, and D and if necessary, round to the nearest tenth of a millimeter. A. Let 'n' be defined as the number of swings the pendulum travels. Define 'a1' and 'r' (these are used in the explicit formula.) Write a geometric sequence in summation (Sigma) notation to express the total distance that the pendulum will have traveled after 30 swings. Sigma notation will have a lower limit, an upper limit, and an explicit formula. Do not compute the sum. B. Use the summation formula to find the total distance the pendulum swings. C. Using information from part A, write a geometric series in summation (Sigma) notation to express the total distance that the pendulum will have travelled when it comes to a complete stop. D. Use the summation formula to find the total distance the pendulum has travelled when it comes to a complete stop. (Hint: this is the sum of an infinite geometric series!)
A museum contains a large pendulum. Due to air resistance and friction, each swing of the pendulum is a little shorter than the previous one. Suppose the first swing of the pendulum has a length of 25,000 mm and each return swing loses 4% of its length until eventually the pendulum comes to a complete stop. Answer parts A, B, C, and D and if necessary, round to the nearest tenth of a millimeter.
A. Let 'n' be defined as the number of swings the pendulum travels. Define 'a1' and 'r' (these are used in the explicit formula.) Write a geometric sequence in summation (Sigma) notation to express the total distance that the pendulum will have traveled after 30 swings. Sigma notation will have a lower limit, an upper limit, and an explicit formula. Do not compute the sum.
B. Use the summation formula to find the total distance the pendulum swings.
C. Using information from part A, write a geometric series in summation (Sigma) notation to express the total distance that the pendulum will have travelled when it comes to a complete stop.
D. Use the summation formula to find the total distance the pendulum has travelled when it comes to a complete stop. (Hint: this is the sum of an infinite geometric series!)
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