document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=621013>Question 621013</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use differences to find a pattern in the sequence.<BR>\n" );
document.write( "-1, -2, -1, 5, 19, 44, 83<BR>\n" );
document.write( "Assuming that the pattern continues, the eight term should be? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #390523 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=390523\">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( "First Differences:  -2 - (-1) = -1, -1 - (-2) = 1, 5 - (-1) = 6, 19 - 5 = 14, 44 - 19 = 25, 83 - 44 = 39\r<BR>\n" );
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document.write( "Second Differences:  1 - (-1) = 2, 6 - 1 = 5, 14 - 6 = 8, 25 - 14 = 11, 39 - 25 = 14\r<BR>\n" );
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document.write( "Third Differences:  5 - 2 = 3, 8 - 5 = 3, 11 - 8 = 3, 14 - 11 = 3\r<BR>\n" );
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document.write( "Third Differences are all the same, therefore we need a cubic polynomial.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ an^3\ +\ bn^2\ + cn\ +\ d\\">\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE n\ =\ 1\">, then the polynomial value is the value of the original first term, namely -1, so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a(1)^3\ +\ b(1)^2\ +\ c(1)\ +\ d\ =\ -1\">\r<BR>\n" );
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document.write( "which is to say:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a\ +\ b\ +\ c\ +\ d\ =\ -1\">\r<BR>\n" );
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document.write( "Similarly:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 8a\ +\ 4b\ +\ 2c\ +\ d\ =\ -2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 27a\ +\ 9b\ +\ 3c\ +\ d\ =\ -1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 64a\ +\ 16b\ +\ 4c\ +\ d\ =\ 5\">\r<BR>\n" );
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document.write( "Solve the 4X4 system of equations by the method of your choice then use the ordered quadruple solution to specify the coefficients of your cubic polynomial.  Finally, calculate the value of the polynomial for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE n\ =\ 8\">\r<BR>\n" );
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document.write( "=====================================================================\r<BR>\n" );
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document.write( "Or you can do it the easy way since you have the first 7 terms and only need the next one.\r<BR>\n" );
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document.write( "If Third Difference number 5 is the same as the others it is 3, then 14 plus 3 is 17 so 17 must be Second Difference number 6.  Then 17 plus 39 is 56 which must be First Difference number 7, so term 7 which is 83 plus 56 is the 8th term, 139.\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<BR>\n" );
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