document.write( "Question 1199440: A ball is dropped from a height of 1.5m. Each time it bounces on the groung, it rises to 3/4 of the height it fell.Calculate the total distance by the 20th bounce. \n" ); document.write( "

Algebra.Com's Answer #833355 by htmentor(1343) $\"\"$ $\"About$

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I interpret \"by the 20th bounce\" to mean up to but not including bounce number 20.
\n" ); document.write( "The height reached by each successive bounce is 3/4 the height of the previous
\n" ); document.write( "bounce. This suggests that we have a geometric series with a common ratio of
\n" ); document.write( "3/4. For all the bounces except the first one, there is a \"round trip\", i.e.
\n" ); document.write( "the ball rises and then falls from the same height. For the 1st bounce, the ball
\n" ); document.write( "starts at a height of 1.5. So our sequence will start with the first return bounce, and we will add the original 1.5 to get the total.
\n" ); document.write( "The distance for the 1st bounce, d_1 = 2*(3/4)*1.5, for the 2nd bounce,
\n" ); document.write( "d_2 = 2*(3/4)*(1.5)*(3.4), etc. So the nth distance will be
\n" ); document.write( "d_n = (9/4)*(3/4)^(n-1). The sum of n terms of a geometric series S_n = a(1-r^n)/(1-r).
\n" ); document.write( "In this case, a = 9/4 and r = 3/4.
\n" ); document.write( "The total distance will be 1.5 + S_18 = 10.45 m.
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