document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209387>Question 1209387</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let<BR>\n" );
document.write( "a + ar + ar^2 + ar^3 + ...<BR>\n" );
document.write( "be an infinite geometric series.  The sum of the series is 3.  The sum of the cubes of all the terms is 5.  Find the common ratio. </SPAN>\n" );
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document.write( "The given series is<br><BR>\n" );
document.write( "a+ar+ar^2+ar^3+...<br><BR>\n" );
document.write( "The sum of the series is 3:<br><BR>\n" );
document.write( "[1] <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2F%281-r%29=3 ALIGN=MIDDLE  ALT=\"a%2F%281-r%29=3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The series consisting of the cubes of the terms of the given series is<br><BR>\n" );
document.write( "a^3r^3+a^3r^6+a^3r^9+...<br><BR>\n" );
document.write( "The sum of that series is 5:<br><BR>\n" );
document.write( "[2] <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E3%2F%281-r%5E3%29=5 ALIGN=MIDDLE  ALT=\"a%5E3%2F%281-r%5E3%29=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "To find the common ratio r, solve [1] for a in terms of r and substitute in [2].<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=3%281-r%29 ALIGN=MIDDLE  ALT=\"a=3%281-r%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%281-r%29%29%5E3%2F%281-r%5E3%29=5 ALIGN=MIDDLE  ALT=\"%283%281-r%29%29%5E3%2F%281-r%5E3%29=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=27%281-3r%2B3r%5E2-r%5E3%29=5%281-r%5E3%29 ALIGN=MIDDLE  ALT=\"27%281-3r%2B3r%5E2-r%5E3%29=5%281-r%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=27-81r%2B81r%5E2-27r%5E3=5-5r%5E3 ALIGN=MIDDLE  ALT=\"27-81r%2B81r%5E2-27r%5E3=5-5r%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=22r%5E3-81r%5E2%2B81r-22=0 ALIGN=MIDDLE  ALT=\"22r%5E3-81r%5E2%2B81r-22=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "By inspection, r=1 is one solution to that equation.  However r=1 produces an infinite geometric series that has no sum.<br><BR>\n" );
document.write( "Use synthetic division to remove the root x=1 to find the other two roots.<br><PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "\r\n" );
document.write( "  1 | 22 -81  81 -22\r\n" );
document.write( "    |     22 -59  22\r\n" );
document.write( "    +---------------\r\n" );
document.write( "      22 -59  22   0</PRE><BR>\n" );
document.write( "The remaining quadratic is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=22x%5E2-59x%2B22 ALIGN=MIDDLE  ALT=\"22x%5E2-59x%2B22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Use the quadratic formula to find that the other two solutions are<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2859%2B-sqrt%281545%29%29%2F44 ALIGN=MIDDLE  ALT=\"%2859%2B-sqrt%281545%29%29%2F44\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "There are two possible values for the common ratio r:<br><BR>\n" );
document.write( "ANSWERS:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F44%29%2859%2Bsqrt%281545%29%29 ALIGN=MIDDLE  ALT=\"%281%2F44%29%2859%2Bsqrt%281545%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F44%29%2859-sqrt%281545%29%29 ALIGN=MIDDLE  ALT=\"%281%2F44%29%2859-sqrt%281545%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );