document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1134409>Question 1134409</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find four numbers in G.P whose sum is 85 and product is 4096. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #751786 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13198)</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=751786\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The four numbers are<br><BR>\n" );
document.write( "a, ar, ar^2, ar^3<br><BR>\n" );
document.write( "Their sum is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%281%2Br%2Br%5E2%2Br%5E3%29 ALIGN=MIDDLE  ALT=\"a%281%2Br%2Br%5E2%2Br%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >; their product is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E4r%5E6 ALIGN=MIDDLE  ALT=\"a%5E4r%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  So<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%281%2Br%2Br%5E2%2Br%5E3%29+=+85 ALIGN=MIDDLE  ALT=\"a%281%2Br%2Br%5E2%2Br%5E3%29+=+85\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E4r%5E6+=+4096 ALIGN=MIDDLE  ALT=\"a%5E4r%5E6+=+4096\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Since both the sum and product are whole numbers, the four numbers are almost certainly whole numbers.  So solve the problem by logical trial and error.<br><BR>\n" );
document.write( "One way is to start with the given product and find the numbers with that product whose sum is 85.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E4r%5E6+=+4096+=+2%5E12 ALIGN=MIDDLE  ALT=\"a%5E4r%5E6+=+4096+=+2%5E12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "A bit of playing around with that finds a=1 and r=2^2=4 is a solution; that gives the four numbers as 1, 4, 16, and 64; and their sum is 85.<br><BR>\n" );
document.write( "Or another way would be to start with the sum of 85 and find four numbers with that sum whose product is 4096.  This one takes a bit more problem solving skill....<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%281%2Br%2Br%5E2%2Br%5E3%29+=+85 ALIGN=MIDDLE  ALT=\"a%281%2Br%2Br%5E2%2Br%5E3%29+=+85\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%281%2Br%29%2Bar%5E2%281%2Br%29+=+85 ALIGN=MIDDLE  ALT=\"a%281%2Br%29%2Bar%5E2%281%2Br%29+=+85\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%281%2Br%29%281%2Br%5E2%29+=+85+=+5%2A17 ALIGN=MIDDLE  ALT=\"a%281%2Br%29%281%2Br%5E2%29+=+85+=+5%2A17\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Again a=1 and r=4 satisfies this equation, because 1+r = 5 and 1+r^2 = 17.<br><BR>\n" );
document.write( "And here is another way to start with the sum of 85 and find the solution by logical reasoning; this path is probably easier to see than the one described above.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%281%2Br%2Br%5E2%2Br%5E3%29+=+85+=+5%2A17 ALIGN=MIDDLE  ALT=\"a%281%2Br%2Br%5E2%2Br%5E3%29+=+85+=+5%2A17\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since the numbers are probably whole numbers, this means a has to be either 1 or 5.  And a bit of playing with those possibilities quickly shows it can't be 5.  So a = 1, and it takes only a bit more trial and error to find that r=4.  Then verifying the solution by showing that the product is 4096 completes the problem. </SPAN>\n" );
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