document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1004512>Question 1004512</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Three numbers whose sum is 3 form an arithmetic sequence and their squares form a geometric sequence. What are the numbers? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #620981 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52788)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=620981\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "Three numbers whose sum is 3 form an arithmetic sequence and their squares form a geometric sequence. <BR>\n" );
document.write( "What are the numbers?<BR>\n" );
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document.write( "<U>Answer</U>. The numbers are &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+-+sqrt%282%29 ALIGN=MIDDLE  ALT=\"1+-+sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%2B+sqrt%282%29 ALIGN=MIDDLE  ALT=\"1+%2B+sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<U>Solution</U>\r<BR>\n" );
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document.write( "Since three numbers form an arithmetic progression, we can represent them as &nbsp;(a-d), &nbsp;a, &nbsp;and &nbsp;(a+d), \r<BR>\n" );
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document.write( "where &nbsp;a &nbsp;is the middle term and &nbsp;d&nbsp; is the common difference.\r<BR>\n" );
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document.write( "The squares of these numbers are &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a-d%29%5E2 ALIGN=MIDDLE  ALT=\"%28a-d%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2 ALIGN=MIDDLE  ALT=\"a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;and &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bd%29%5E2 ALIGN=MIDDLE  ALT=\"%28a%2Bd%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "The fact that the squares form a geometric progression means that \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28a%2Bd%29%5E2%29%2Fa%5E2 ALIGN=MIDDLE  ALT=\"%28%28a%2Bd%29%5E2%29%2Fa%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2F%28%28a-d%29%5E2%29 ALIGN=MIDDLE  ALT=\"a%5E2%2F%28%28a-d%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > &nbsp;&nbsp;&nbsp;&nbsp;(1)\r<BR>\n" );
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document.write( "(the ratio of the third to the second is equal to the ratio of the second to the first, as these ratios are \r<BR>\n" );
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document.write( "the common ratio of the geometric progression). &nbsp;The formula &nbsp;(1) implies, &nbsp;after simplifying, that\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5E2 ALIGN=MIDDLE  ALT=\"d%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%5E2 ALIGN=MIDDLE  ALT=\"2a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;&nbsp;&nbsp;&nbsp;or &nbsp;&nbsp;&nbsp;&nbsp;d = a<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Next, &nbsp;since the sum of the tree numbers is &nbsp;3, &nbsp;we conclude that\r<BR>\n" );
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document.write( "(a-d) + a + (a+d) = 3a = 3, &nbsp;&nbsp;and, &nbsp;&nbsp;hence, &nbsp;&nbsp;a = 1. &nbsp;&nbsp;In turn, it means that &nbsp;d = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Thus our sequence of three numbers is &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+-+sqrt%282%29 ALIGN=MIDDLE  ALT=\"1+-+sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;1, &nbsp;and &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%2B+sqrt%282%29 ALIGN=MIDDLE  ALT=\"1+%2B+sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "As a check, &nbsp;it is clear that the found numbers form arithmetic progression and their sum is &nbsp;3.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Their squares are &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3-2sqrt%282%29 ALIGN=MIDDLE  ALT=\"3-2sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;1, &nbsp;and &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2B2sqrt%282%29 ALIGN=MIDDLE  ALT=\"3%2B2sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The consequtive ratios of the squares are &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%283-2sqrt%282%29%29 ALIGN=MIDDLE  ALT=\"1%2F%283-2sqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > &nbsp;&nbsp;and &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2B2sqrt%282%29%29%2F1 ALIGN=MIDDLE  ALT=\"%283%2B2sqrt%282%29%29%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Finally, &nbsp;you can check yourself that these ratios are equal, &nbsp;which means that the squares form \r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the geometric progression. \r<BR>\n" );
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document.write( "The solution is completed.\r<BR>\n" );
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document.write( "Thank you for submitting the fresh, &nbsp;sweet and crispy problem!\r<BR>\n" );
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document.write( " </SPAN>\n" );
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