document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=643336>Question 643336</A>: <I><SPAN STYLE=\"word-break: break-all;\">  What is the pattern for the following sequence Term 1,2,3,4,5, n Value 0,3,7,12,18 ? etc. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #404452 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=404452\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Use the process of successive differences.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "3 minus 0 is 3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "7 minus 3 is 4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "12 minus 7 is 5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "18 minus 12 is 6\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "4 minus 3 is 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "5 minus 4 is 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "6 minus 5 is 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since all of the differences are the same after two passes, you know a quadratic will suffice for a rule (one pass: linear, three passes: cubic, etc.)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So you need a function of the form <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f(n)\ =\ an^2\ +\ bn\ +\ c\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Plugging in numbers you get\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a(1)^2\ +\ b(1)\ +\ c\ =\ 0\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a(2)^2\ +\ b(2)\ +\ c\ =\ 3\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a(3)^2\ +\ b(3)\ +\ c\ =\ 7\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Which leads to the following 3X3 system:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a\ +\ b\ +\ c\ =\ 0\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4a\ +\ 2b\ +\ c\ =\ 3\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 9a\ +\ 3b\ +\ c\ =\ 7\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Which you can solve by your favorite process to get the coefficients of your function.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There is actually a shortcut for sequences.  The lead coefficient of your function is always one-half of the value of the value of the constant difference.  The constant difference here was 1, so the lead coefficient is 1/2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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" );
document.write( "<div  DISABLED_style= \"text-align:center\"><a  rel=nofollow HREF=\"http://outcampaign.org/\" target=\"_blank\"><IMG STYLE=\"max-width:80%;\" src=\"http://cdn.cloudfiles.mosso.com/c116811/scarlet_A.png\" border=\"0\" alt=\"The Out Campaign: Scarlet Letter of Atheism\" width=\"143\" height=\"122\" /></a></div><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );