document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=453965>Question 453965</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use first and second differences to identify the next three terms in the following sequence: 12, 25, 44, 69, 100, 137, 180,... \r<BR>\n" );
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document.write( "a)	180, 229, 284<BR>\n" );
document.write( "b)	180, 274, 335<BR>\n" );
document.write( "c)	229, 284, 345<BR>\n" );
document.write( "d)	229, 345, 412 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #311814 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=311814\">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:\r<BR>\n" );
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document.write( "13, 19, 25, 31, 37, 43\r<BR>\n" );
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document.write( "Second differences:\r<BR>\n" );
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document.write( "6, 6, 6, 6, 6\r<BR>\n" );
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document.write( "Since the second differences are all equal, the series can be modeled with a 2nd degree polynomial.\r<BR>\n" );
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document.write( "Consider the 2nd degree (quadratic) polynomial equation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ an^2\ +\ bn\ +\ c\ =\ A_n\">\r<BR>\n" );
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document.write( "Where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\"> is the ordinal of the term and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large A_n\"> is the value of the term.\r<BR>\n" );
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document.write( "The value of the 1st term is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large A_1\ =\ 12\">, therefore:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a(1)^2\ +\ b(1)\ +\ c\ =\ 12\">\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\ =\ 12\">\r<BR>\n" );
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document.write( "Similarly, using the 2nd and 3rd terms we can derive:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4a\ +\ 2b\ +\ c\ =\ 25\">\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 \ \ \ \ \ \ \ \ \ \ 9a\ +\ 3b\ +\ c\ =\ 44\">\r<BR>\n" );
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document.write( "Solve the 3 by 3 system of linear equations using any desired method (I use the MDETERM function and Cramer's Rule in Excel to solve anything more complex than a simple 2X2) to calculate the coefficients of your quadratic model for your sequence.  Then substitute 8, 9, and 10 for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\"> in the model to develop the 8th, 9th, and 10th terms of the 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<BR>\n" );
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