document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=995748>Question 995748</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 3 6 17 42 87 158 find the sequence <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #614410 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=614410\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "Find the common difference:<BR>\n" );
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document.write( "   3     6     17     42     87    158\r\n" );
document.write( "    \   / \    / \    / \    / \    / \r\n" );
document.write( "      3     11     25     45     71\r\n" );
document.write( "       \    / \    / \    / \    / \r\n" );
document.write( "          8     14     20     26\r\n" );
document.write( "           \    / \    / \    /\r\n" );
document.write( "              6      6      6\r\n" );
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document.write( "The actual difference numbers don't matter.  The important thing is that the differences are the same in the third row of differences.  Hence, the sequence can be modeled with a 3rd-degree polynomial.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ S(n)\ =\ an^3\ +\ bn^2\ +\ cn\ +\ d\">\r<BR>\n" );
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document.write( "The value of the function at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\ =\ 1\"> is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 3\">,\r<BR>\n" );
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document.write( "Hence\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ S(1)\ =\ a(1)^3\ +\ b(1)^2\ +\ c(1)\ +\ d\ =\ 3\">\r<BR>\n" );
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document.write( "Or, more simply put\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a\ +\ b\ +\ c\ +\ d\ =\ 3\">\r<BR>\n" );
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document.write( "Then, when <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\ =\ 2\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large S(n)\ =\ 6\">.  Similar to above we derive:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 8a\ +\ 4b\ +\ 2c\ +\ d\ =\ 6\">\r<BR>\n" );
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document.write( "In a like manner we can derive:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 27a\ +\ 9b\ +\ 3c\ +\ d\ =\ 17\">\r<BR>\n" );
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document.write( "And\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 64a\ +\ 16b\ +\ 4c\ +\ d\ =\ 42\">\r<BR>\n" );
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document.write( "Solve the 4X4 system of linear equations to find the coefficients of the 3rd-degree polynomial that models your desired sequence.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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