document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1021597>Question 1021597</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello, this was in one of my sequence/series questions.\r<BR>\n" );
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document.write( "Im pretty sure its got something to do with geometric series, but i dont know how to problem solve using the formula.\r<BR>\n" );
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document.write( "Please help!\r<BR>\n" );
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document.write( "A bouncy ball is dropped from a height of h meters and allowed to bounce indefinitely. Each succeeding bounce is 0.8 of the preceding one...<BR>\n" );
document.write( "If the total vertical distance traveled by the ball is 63 meters, from what height was the ball dropped?\r<BR>\n" );
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document.write( "Thank you very much </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #637348 by </B><A HREF=/tutors/aboutme.mpl?userid=FrankM><B>FrankM(1040)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=FrankM><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=FrankM><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=637348\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The infinite series summation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D%281%2F%281-r%29%29 ALIGN=MIDDLE  ALT=\"a%5B1%5D%281%2F%281-r%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the first term of the series, and r is the geometric multiplier. \r<BR>\n" );
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document.write( "Say you were given 1+1/2+1/4+1/8+..... can you see that this adds to 2? Take 2 squares, the first is \"1\" of course then take the second and cut it in half with a line, then the remaining half to quarters. It reproduces the series, and from the equation -\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%281%2F%281-.5%29%29=2 ALIGN=MIDDLE  ALT=\"1%281%2F%281-.5%29%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now, your problem is tricky as there are 2 series to add, H(1+.8+.64+.512.....) this is vertical distance dropped. But you need to add the distance it goes up, H(.8+.64+.512.....) see how the first \"up\" is .8 the height of the drop? \r<BR>\n" );
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document.write( "Let's solve the first series -\r<BR>\n" );
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document.write( "1+.8+.64+.512...    (we'll multiply by H after solving this)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D=1 ALIGN=MIDDLE  ALT=\"a%5B1%5D=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and r=.8\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%281%2F%281-.8%29%29=5 ALIGN=MIDDLE  ALT=\"1%281%2F%281-.8%29%29=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the first series adds to 5H\r<BR>\n" );
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document.write( "The second series \r<BR>\n" );
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document.write( ".8+.64+.512..\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D=.8 ALIGN=MIDDLE  ALT=\"a%5B1%5D=.8\"   DISABLED_style= \"max-width:90%; height:auto;\" > and r=.8\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=.8%281%2F%281-.8%29%29=4 ALIGN=MIDDLE  ALT=\".8%281%2F%281-.8%29%29=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "So this adds to 4H which makes sense, as the \"up\" is .8 times the prior \"down\".\r<BR>\n" );
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document.write( "Now, you've calculated that from a drop of H and a boucyness of .8, you get a total travel of 9H. <BR>\n" );
document.write( "One final step. They told you the travel was 63 meters.\r<BR>\n" );
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document.write( "9H = 63m\r<BR>\n" );
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document.write( "H=63/9 m\r<BR>\n" );
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document.write( "H= 7m \r<BR>\n" );
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document.write( "Now you can solve an infinite number of infinite geometric series equations..... \r<BR>\n" );
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document.write( "NOTE: The tutors here have no way to communicate with each other. I believe the other 2 answers are wrong.  Stanbon did not include the \"up\" distance. The problem states \"total vertical distance traveled\" .<BR>\n" );
document.write( "Roth started at .8h which ignored the initial drop, the first full 1h. (Edit again - Roth has corrected and confirmed my answer) </SPAN>\n" );
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