document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=933337>Question 933337</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For each sequence, find a closed formula for the general term, a sub n.<BR>\n" );
document.write( "1. 53,477,4293,38637,347733,... a sub n =<BR>\n" );
document.write( "2. 2,5,10,17,26,...a sub n =<BR>\n" );
document.write( "3. -2,-8,-18,-32,-50,...a sub n =<BR>\n" );
document.write( "Help would truly be appreciated! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #566768 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=566768\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a sub n means the \"nth\" term of the sequence.\r<BR>\n" );
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document.write( "Example: 1,2,3,4,5,....,\r<BR>\n" );
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document.write( "the nth term is n. We say \"a sub n = n\"\r<BR>\n" );
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document.write( "Example 2: 1,4,9,16,25,...\r<BR>\n" );
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document.write( "We say \"a sub n = n^2\".\r<BR>\n" );
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document.write( "1. 53,477,4293,38637,347733,...\r<BR>\n" );
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document.write( "to get next term multiply previous one by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " a sub n =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=53%2A9%5En ALIGN=MIDDLE  ALT=\"a%5Bn%5D=53%2A9%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" >(for all terms given)<BR>\n" );
document.write( " if n=0,1,2...\r<BR>\n" );
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document.write( "check:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B0%5D=53%2A9%5E0=53 ALIGN=MIDDLE  ALT=\"a%5B0%5D=53%2A9%5E0=53\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D=53%2A9%5E1=477 ALIGN=MIDDLE  ALT=\"a%5B1%5D=53%2A9%5E1=477\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B2%5D=53%2A9%5E2=4293 ALIGN=MIDDLE  ALT=\"a%5B2%5D=53%2A9%5E2=4293\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B3%5D=53%2A9%5E3=38637 ALIGN=MIDDLE  ALT=\"a%5B3%5D=53%2A9%5E3=38637\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B4%5D=53%2A9%5E4=347733 ALIGN=MIDDLE  ALT=\"a%5B4%5D=53%2A9%5E4=347733\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D+=+53+%2A9%5E%28n-1%29 ALIGN=MIDDLE  ALT=\"a%5Bn%5D+=+53+%2A9%5E%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > if n=1,2... \r<BR>\n" );
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document.write( "check:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D+=+53+%2A9%5E%281-1%29=53%2A9%5E0=53%2A1=53 ALIGN=MIDDLE  ALT=\"a%5B1%5D+=+53+%2A9%5E%281-1%29=53%2A9%5E0=53%2A1=53\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( ":\r<BR>\n" );
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document.write( "I prefer this one because is better to have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > as first term\r<BR>\n" );
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document.write( "2. 2,5,10,17,26,...<BR>\n" );
document.write( "pattern\r<BR>\n" );
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document.write( "2----  5----  10----  17----  26,..<BR>\n" );
document.write( "--   3----  5----    7--   9...............first diff\r<BR>\n" );
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document.write( "----     2----   2----   2................second diff.; all <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >s, so degree is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  half of it (or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)will be coefficient of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E2 ALIGN=MIDDLE  ALT=\"n%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "use<BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=an%5E2+%2B+bn+%2B+c ALIGN=MIDDLE  ALT=\"an%5E2+%2B+bn+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" > , set up system of three equations to find coefficients <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" >   \r<BR>\n" );
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document.write( "For instance, I know that the first term (that is, the term when n = 1) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so I'll plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the value:\r<BR>\n" );
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document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%281%29%5E2+%2B+b%281%29+%2B+c+=+a+%2B+b+%2B+c+=+2 ALIGN=MIDDLE  ALT=\"a%281%29%5E2+%2B+b%281%29+%2B+c+=+a+%2B+b+%2B+c+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The second term (that is, the term when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+2 ALIGN=MIDDLE  ALT=\"n+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is 5, so:\r<BR>\n" );
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document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%282%29%5E2+%2B+b%282%29+%2B+c+=+4a+%2B+2b+%2B+c+=+5 ALIGN=MIDDLE  ALT=\"a%282%29%5E2+%2B+b%282%29+%2B+c+=+4a+%2B+2b+%2B+c+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The third term (that is, the term when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+3 ALIGN=MIDDLE  ALT=\"n+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so:\r<BR>\n" );
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document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%283%29%5E2+%2B+b%283%29+%2B+c+=+9a+%2B+3b+%2B+c+=+10 ALIGN=MIDDLE  ALT=\"a%283%29%5E2+%2B+b%283%29+%2B+c+=+9a+%2B+3b+%2B+c+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%2B+b+%2B+c+=+2 ALIGN=MIDDLE  ALT=\"a+%2B+b+%2B+c+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.........1<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a+%2B+2b+%2B+c+=+5 ALIGN=MIDDLE  ALT=\"4a+%2B+2b+%2B+c+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" >...........2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9a+%2B+3b+%2B+c+=+10 ALIGN=MIDDLE  ALT=\"9a+%2B+3b+%2B+c+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" >...........3<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%2B+b+%2B+c+=+2 ALIGN=MIDDLE  ALT=\"a+%2B+b+%2B+c+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.........1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a++=+2-+b+-c ALIGN=MIDDLE  ALT=\"a++=+2-+b+-c\"   DISABLED_style= \"max-width:90%; height:auto;\" >...plug in 2..........................a\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%282-+b+-c%29%2B+2b+%2B+c+=+5 ALIGN=MIDDLE  ALT=\"4%282-+b+-c%29%2B+2b+%2B+c+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" >...........2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8-+4b+-4c%2B+2b+%2B+c+=+5 ALIGN=MIDDLE  ALT=\"8-+4b+-4c%2B+2b+%2B+c+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8-+2b+-3c=+5 ALIGN=MIDDLE  ALT=\"8-+2b+-3c=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8-+5-3c=+2b ALIGN=MIDDLE  ALT=\"8-+5-3c=+2b\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3-3c=+2b ALIGN=MIDDLE  ALT=\"3-3c=+2b\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F2-3c%2F2=+b ALIGN=MIDDLE  ALT=\"3%2F2-3c%2F2=+b\"   DISABLED_style= \"max-width:90%; height:auto;\" > ...........................b\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%282-+b+-c%29+%2B+3b+%2B+c+=+10 ALIGN=MIDDLE  ALT=\"9%282-+b+-c%29+%2B+3b+%2B+c+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" >...........3...plug in a\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18-9+b+-9c%2B+3b+%2B+c+=+10 ALIGN=MIDDLE  ALT=\"18-9+b+-9c%2B+3b+%2B+c+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18-6+b+-8c++=+10 ALIGN=MIDDLE  ALT=\"18-6+b+-8c++=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" >......plug in b\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18-6%283%2F2-3c%2F2%29-8c+=+10-18 ALIGN=MIDDLE  ALT=\"18-6%283%2F2-3c%2F2%29-8c+=+10-18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9%2B9c+-8c+=+-8 ALIGN=MIDDLE  ALT=\"-9%2B9c+-8c+=+-8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c++=+-8%2B9 ALIGN=MIDDLE  ALT=\"c++=+-8%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c++=1 ALIGN=MIDDLE  ALT=\"c++=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F2-3c%2F2=+b+ ALIGN=MIDDLE  ALT=\"3%2F2-3c%2F2=+b+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F2-3%2A1%2F2=+b+ ALIGN=MIDDLE  ALT=\"3%2F2-3%2A1%2F2=+b+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28b=0+%29 ALIGN=MIDDLE  ALT=\"highlight%28b=0+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a++=+2-+b+-c ALIGN=MIDDLE  ALT=\"a++=+2-+b+-c\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a++=+2-+0+-1 ALIGN=MIDDLE  ALT=\"a++=+2-+0+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28a=1%29 ALIGN=MIDDLE  ALT=\"highlight%28a=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so, we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+1 ALIGN=MIDDLE  ALT=\"a+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+0 ALIGN=MIDDLE  ALT=\"b+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+1 ALIGN=MIDDLE  ALT=\"c+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the formula is:\r<BR>\n" );
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document.write( "    1n^2 + 0n + 1 = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E2+%2B+1 ALIGN=MIDDLE  ALT=\"n%5E2+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so, a sub n =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E2%2B1 ALIGN=MIDDLE  ALT=\"n%5E2%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >(for all terms given)\r<BR>\n" );
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document.write( "3. -2,-8,-18,-32,-50,...a sub n =\r<BR>\n" );
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document.write( "-2 ----  -8 ----  -18----   -32 ----  -50\r<BR>\n" );
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document.write( "----..-6----..-10----..-14---..-18\r<BR>\n" );
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document.write( "----       .. -4----   .. -4----  ..  -4.........third difference, -> 3rd degree; constant is -4, so half of it (or -2)will be coefficient of n^2 \r<BR>\n" );
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document.write( "a sub n = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2n%5E2+ ALIGN=MIDDLE  ALT=\"-2n%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >(for all terms given)<BR>\n" );
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