document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1100175>Question 1100175</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Fimd six numbers in AP, such that the sum of the two extremes be 16 and the product of the two middle terms be 63.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #714745 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=714745\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br>If the sum of the first and last numbers is an integer and the product of the two middle terms is an integer, then almost certainly all the numbers will be integers.<br><BR>\n" );
document.write( "Then if the product of the middle two numbers is 63, the two middle numbers are 7 and 9.<br><BR>\n" );
document.write( "So the common difference is 2; that makes the 6 numbers<br><BR>\n" );
document.write( "3, 5, 7, 9, 11, 13<br><br><BR>\n" );
document.write( "You could, of course, solve the problem using algebra; however, it takes far more work than the logical analysis solution shown above.<br><BR>\n" );
document.write( "Let the first term be a and the common difference be d; then the 6 numbers are<BR>\n" );
document.write( "a, a+d, a+2d, a+3d, a+4d, a+5d<br><BR>\n" );
document.write( "Then<BR>\n" );
document.write( "(1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%2B5d+=+16 ALIGN=MIDDLE  ALT=\"2a%2B5d+=+16\"   DISABLED_style= \"max-width:90%; height:auto;\" >  the sum of the first and last numbers is 16<BR>\n" );
document.write( "(2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2B2d%29%28a%2B3d%29+=+63 ALIGN=MIDDLE  ALT=\"%28a%2B2d%29%28a%2B3d%29+=+63\"   DISABLED_style= \"max-width:90%; height:auto;\" >  the product of the two middle numbers is 63<br><BR>\n" );
document.write( "Solve equation (1) for a: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+%2816-5d%29%2F2 ALIGN=MIDDLE  ALT=\"a+=+%2816-5d%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Substitute into equation (2):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%2816-d%29%2F2%29%28%2816%2Bd%29%2F2%29+=+63 ALIGN=MIDDLE  ALT=\"%28%2816-d%29%2F2%29%28%2816%2Bd%29%2F2%29+=+63\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28256-d%5E2%29%2F4+=+63 ALIGN=MIDDLE  ALT=\"%28256-d%5E2%29%2F4+=+63\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=256-d%5E2+=+252 ALIGN=MIDDLE  ALT=\"256-d%5E2+=+252\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5E2+=+4 ALIGN=MIDDLE  ALT=\"d%5E2+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+2 ALIGN=MIDDLE  ALT=\"d+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Substitute back into equation (1) to find a:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%2B10+=+16 ALIGN=MIDDLE  ALT=\"2a%2B10+=+16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a+=+6 ALIGN=MIDDLE  ALT=\"2a+=+6\"   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 ALIGN=MIDDLE  ALT=\"a+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br><BR>\n" );
document.write( "I think I like the logical analysis solution better....<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );