document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=847871>Question 847871</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the formula for this sequence:<BR>\n" );
document.write( " 21, 45, 77, 117, 165 ?<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 #510715 by </B><A HREF=/tutors/aboutme.mpl?userid=swincher4391><B>swincher4391(1107)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=swincher4391><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=swincher4391><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=510715\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's look at some things. What is the difference between these numbers?\r<BR>\n" );
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document.write( "45 -21 = 24\r<BR>\n" );
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document.write( "77-45 = 32\r<BR>\n" );
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document.write( "117 - 77 = 40...\r<BR>\n" );
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document.write( "etc.  It seems as though our difference is increasing by 8 each time.\r<BR>\n" );
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document.write( "By knowing that the 2nd difference is 8, we know that the function is of the form 4n^2 + something. In general, an^2 has second difference 2a.\r<BR>\n" );
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document.write( "We also realize that the difference between a_0 and a_1 is 16.\r<BR>\n" );
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document.write( "So a_0 = 5.\r<BR>\n" );
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document.write( "So we have 4n^2 + b*n + 5\r<BR>\n" );
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document.write( "This should be enough to figure out what b is.\r<BR>\n" );
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document.write( "Take a_2 for instance.\r<BR>\n" );
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document.write( "4(2)^2 + b*2 + 5  = 45\r<BR>\n" );
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document.write( "16 + 2b + 5 = 45\r<BR>\n" );
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document.write( "2b = 24\r<BR>\n" );
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document.write( "b = 12\r<BR>\n" );
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document.write( "So our entire function a_n  =  4n^2 + 12n + 5\r<BR>\n" );
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document.write( "We check our answer:\r<BR>\n" );
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document.write( "a(1) =  4 + 12 + 5 = 21\r<BR>\n" );
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document.write( "a(2) =  4*4 + 24 + 5 = 45\r<BR>\n" );
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document.write( "a(3) =  4*9 + 36 + 5 = 77\r<BR>\n" );
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document.write( "a(4) =  4*16 + 48 + 5 = 117\r<BR>\n" );
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document.write( "a(5) =  4*25 + 60 + 5 =  165\r<BR>\n" );
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document.write( "Just for fun a(10) = 4*100 + 12*10 + 5 =  525.  Simple.<BR>\n" );
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