document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=902538>Question 902538</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If x = -1, find the value of 2013x^2013 + 2012x^2012 + 2011x^2011 + 2010x^2010 + . . . + 2x^2 + x.\r<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 #547397 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=547397\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i'm not sure if this is the way you're supposed to solve this, but i do believe i did solve it.\r<BR>\n" );
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document.write( "it doesn't look like an arithmetic series or a geometric series so those formulas don't appear to work with this.\r<BR>\n" );
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document.write( "the first term is x<BR>\n" );
document.write( "the second term is 2x^2<BR>\n" );
document.write( "the third term is presumably 3x^3<BR>\n" );
document.write( "the fourth term is presumably 4x^4<BR>\n" );
document.write( "the 2013th term is 2013x^2013 as shown.\r<BR>\n" );
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document.write( "when x = -1, this turns out to be:\r<BR>\n" );
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document.write( "first term is -1<BR>\n" );
document.write( "second term is 2<BR>\n" );
document.write( "third term is -3<BR>\n" );
document.write( "fourth term is 4<BR>\n" );
document.write( "etc.\r<BR>\n" );
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document.write( "i then looked for a pattern.<BR>\n" );
document.write( "the pattern i found is shown here.\r<BR>\n" );
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document.write( "n	An	Sn\r\n" );
document.write( "1	-1	-1\r\n" );
document.write( "2	2	1\r\n" );
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document.write( "3	-3	-2\r\n" );
document.write( "4	4	2\r\n" );
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document.write( "5	-5	-3\r\n" );
document.write( "6	6	3\r\n" );
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document.write( "7	-7	-4\r\n" );
document.write( "8	8	4\r\n" );
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document.write( "9	-9	-5\r\n" );
document.write( "10	10	5\r\n" );
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document.write( "11	-11	-6\r\n" );
document.write( "12	12	6\r\n" );
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document.write( "13	-13	-7\r\n" );
document.write( "14	14	7\r\n" );
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document.write( "15	-15	-8\r\n" );
document.write( "16	16	8\r\n" );
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document.write( "17	-17	-9\r\n" );
document.write( "18	18	9\r\n" );
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document.write( "19	-19	-10\r\n" );
document.write( "20	20	10\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "n is the term in the series.<BR>\n" );
document.write( "An is the nth term in the series.<BR>\n" );
document.write( "Sn is the sum of the n terms in the series.\r<BR>\n" );
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document.write( "if you look at the series, you will see that the sum of n terms, where n is even, is equal to exactly one half of the nth term.\r<BR>\n" );
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document.write( "the sum of 16 terms is 8<BR>\n" );
document.write( "the sum of 18 terms is 9<BR>\n" );
document.write( "the sum of 20 terms is 10\r<BR>\n" );
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document.write( "if you look at the sum of the odd terms, you will see that the sum of the n terms, where n is odd, is equal to the negative of the sum of the n+1 terms.\r<BR>\n" );
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document.write( "the sum of 15 terms is -8<BR>\n" );
document.write( "the sum of 17 terms is -9<BR>\n" );
document.write( "the sum of 19 terms is -10\r<BR>\n" );
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document.write( "you can extrapolate from this to get the sum of 2013 terms as follows.\r<BR>\n" );
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document.write( "the sum of 2014 terms is 1007<BR>\n" );
document.write( "therefore the sum of 2013 terms is -1007.\r<BR>\n" );
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document.write( "that should be your answer.\r<BR>\n" );
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document.write( "i confirmed using excel that this is the case so i'm reasonably confident this is the answer you are looking for unless i completely misinterpreted the problem.\r<BR>\n" );
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