document.write( "Question 1078519: HELP ME??\r
\n" ); document.write( "\n" ); document.write( "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. \r
\n" ); document.write( "\n" ); document.write( "a) On which swing will the length first have a length less than 50 cm?
\n" ); document.write( "b) Find the total distance traveled by the pendulum until it comes to rest. \n" ); document.write( "
Algebra.Com's Answer #692953 by htmentor(1343)\"\" \"About 
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On the first swing, the distance traveled is 100, and on the second swing, the distance is 99.
\n" ); document.write( "The first term of the geometric sequence is, a1 = 100
\n" ); document.write( "The second term of the sequence is, a2 = 99
\n" ); document.write( "The common ratio r = 99/100
\n" ); document.write( "Therefore, the expression for the n-th term of the sequence is:
\n" ); document.write( "a_n = 100*(99/100)^(n-1)
\n" ); document.write( "(a) To answer a, we need to find the value of n for which a_n < 50:
\n" ); document.write( "50 = 100*(99/100)^(n-1)
\n" ); document.write( "log(1/2)/log(99/100) = n - 1
\n" ); document.write( "This gives n = 69.97, and rounding up to the next integer gives n = 70.
\n" ); document.write( "(b) Theoretically, using this formula, it would take an infinite number of trips to come to rest.
\n" ); document.write( "Hence we need to find the sum of the series a_n = 100*(99/100)^(n-1) from n=1 to n=infinity.
\n" ); document.write( "The sum of an infinite series is given by S = a/(1-r) where a=the first term and r=the common ratio.
\n" ); document.write( "Therefore, the total distance traveled is S = 100/(1-(99/100) = 100/0.01 = 10,000 cm.
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