Question 1078519: HELP ME??
Because of the friction and air resistance, each swing of a pendulum is a little shorter than the previous one. The lengths of the swings form a geometric sequence. Suppose the first swing on the pendulum has an arc length of 100cm and a return swing of 99cm.
a) On which swing will the length first have a length less than 50 cm?
b) Find the total distance traveled by the pendulum until it comes to rest.
Answer by htmentor(1343)   (Show Source):
You can put this solution on YOUR website! On the first swing, the distance traveled is 100, and on the second swing, the distance is 99.
The first term of the geometric sequence is, a1 = 100
The second term of the sequence is, a2 = 99
The common ratio r = 99/100
Therefore, the expression for the n-th term of the sequence is:
a_n = 100*(99/100)^(n-1)
(a) To answer a, we need to find the value of n for which a_n < 50:
50 = 100*(99/100)^(n-1)
log(1/2)/log(99/100) = n - 1
This gives n = 69.97, and rounding up to the next integer gives n = 70.
(b) Theoretically, using this formula, it would take an infinite number of trips to come to rest.
Hence we need to find the sum of the series a_n = 100*(99/100)^(n-1) from n=1 to n=infinity.
The sum of an infinite series is given by S = a/(1-r) where a=the first term and r=the common ratio.
Therefore, the total distance traveled is S = 100/(1-(99/100) = 100/0.01 = 10,000 cm.
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