document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1122514>Question 1122514</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The sum of an infinite G.P. is 15 and the sum of the squares of the terms is 45. Find the G.P </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #738653 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=738653\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Let the GP be<br><BR>\n" );
document.write( "a, ar, ar^2, ar^3, ...<br><BR>\n" );
document.write( "Then the GP with the terms squared is<br><BR>\n" );
document.write( "a^2, a^2r^2, a^2r^4, a^2r^6, ...<br><BR>\n" );
document.write( "The infinite sum of the GP is<br><BR>\n" );
document.write( "(1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2F%281-r%29+=+15 ALIGN=MIDDLE  ALT=\"a%2F%281-r%29+=+15\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The infinite sum of the squared GP is<br><BR>\n" );
document.write( "(2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2F%281-r%5E2%29+=+45 ALIGN=MIDDLE  ALT=\"a%5E2%2F%281-r%5E2%29+=+45\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Dividing (2) by (1) gives us<br><BR>\n" );
document.write( "(3) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2F%281%2Br%29+=+3 ALIGN=MIDDLE  ALT=\"a%2F%281%2Br%29+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Then (1) and (3) give us two linear equations in a and r that we can solve:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+15-15r ALIGN=MIDDLE  ALT=\"a+=+15-15r\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+3%2B3r ALIGN=MIDDLE  ALT=\"a+=+3%2B3r\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15-15r+=+3%2B3r ALIGN=MIDDLE  ALT=\"15-15r+=+3%2B3r\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12+=+18r ALIGN=MIDDLE  ALT=\"12+=+18r\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(4) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+2%2F3 ALIGN=MIDDLE  ALT=\"r+=+2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Then substituting (4) in (1) gives us a=5.<br><BR>\n" );
document.write( "The GP has first term 5 and common ratio 2/3:<br><BR>\n" );
document.write( "5, 10/3, 20/9, ...<br><BR>\n" );
document.write( "The squared GP is<br><BR>\n" );
document.write( "25, 100/9 400/81, ...<br><BR>\n" );
document.write( "CHECK:<br><BR>\n" );
document.write( "The sum of the GP is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F%281-%282%2F3%29%29+=+5%2F%281%2F3%29+=+15%29 ALIGN=MIDDLE  ALT=\"5%2F%281-%282%2F3%29%29+=+5%2F%281%2F3%29+=+15%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The sum of the squared GP is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25%2F%281-%284%2F9%29%29+=+25%2F%285%2F9%29+=+45 ALIGN=MIDDLE  ALT=\"25%2F%281-%284%2F9%29%29+=+25%2F%285%2F9%29+=+45\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );