document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209331>Question 1209331</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi<BR>\n" );
document.write( "Find the value of 1-2+3-4+5-6+.....+2007-2008+2009-2010+2011. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848384 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848384\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font color=red>1006</font>\r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "Tutor ikleyn has shown an efficient pathway. Perhaps the most efficient route. <BR>\n" );
document.write( "I'll show an alternative method.\r<BR>\n" );
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document.write( "Let's place the terms of 1-2+3-4+5-6+...+2007-2008+2009-2010+2011 into a table\r<BR>\n" );
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document.write( "<tr><TD >1</TD><TD >-2</TD><TD >3</TD><TD >-4</TD><TD >5</TD><TD >-6</TD><TD >...</TD><TD >-2006</TD><TD >2007</TD><TD >-2008</TD><TD >2009</TD><TD >-2010</TD><TD >2011</TD></tr>\n" );
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document.write( "Then we'll make a copy of this row and reverse it to place under the current terms<BR>\n" );
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document.write( "<tr><TD >1</TD><TD >-2</TD><TD >3</TD><TD >-4</TD><TD >5</TD><TD >-6</TD><TD >...</TD><TD >-2006</TD><TD >2007</TD><TD >-2008</TD><TD >2009</TD><TD >-2010</TD><TD >2011</TD></tr><tr><TD >2011</TD><TD >-2010</TD><TD >2009</TD><TD >-2008</TD><TD >2007</TD><TD >-2006</TD><TD >...</TD><TD >-6</TD><TD >5</TD><TD >-4</TD><TD >3</TD><TD >-2</TD><TD >1</TD></tr>\n" );
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document.write( "Add straight down to see each column adds to either 2012 or -2012.<BR>\n" );
document.write( "More specifically the odd values add to 2012 while the even values add to -2012.\r<BR>\n" );
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document.write( "In the set {1,2,3,...,2010,2011} there are 2010/2 = 1005 even numbers. <BR>\n" );
document.write( "Those even numbers are 2,4,6,...,2010.<BR>\n" );
document.write( "This means that we have 1005 instances of the sum -2012 show up.<BR>\n" );
document.write( "The other 2011-1005 = 1006 sums are 2012\r<BR>\n" );
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document.write( "We then have<BR>\n" );
document.write( "1006*2012 + 1005*(-2012)<BR>\n" );
document.write( "= 2012*(1006-1005)<BR>\n" );
document.write( "= 2012*(1)<BR>\n" );
document.write( "= 2012\r<BR>\n" );
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document.write( "This is not the final answer. <BR>\n" );
document.write( "It would be nice if 1-2+3-4+...+2009-2010+2011 did evaluate to 2012. \r<BR>\n" );
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document.write( "However, when I made that 2nd table, where the bottom row is the reverse of the top row, I introduced a second copy of the sum. Thereby the result of adding everything in that 2nd table would be 2*S, where S = 1-2+3-4+...+2009-2010+2011\r<BR>\n" );
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document.write( "So we have to divide by 2 to correct this error.<BR>\n" );
document.write( "2012/2 = <font color=red>1006</font> is the final answer.\r<BR>\n" );
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document.write( "In other words, S = 1-2+3-4+...+2009-2010+2011 represents combining terms along the top row<BR>\n" );
document.write( "S = 2011-2010+2009-...-4+3-2+1 also happens when we combine terms along the bottom row.<BR>\n" );
document.write( "Add those equations straight down to arrive at 2S = 2012 which leads to S = <font color=red>1006</font><BR>\n" );
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