Question 1209331
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Answer: <font color=red>1006</font>


Explanation


Tutor ikleyn has shown an efficient pathway. Perhaps the most efficient route. 
I'll show an alternative method.


Let's place the terms of 1-2+3-4+5-6+...+2007-2008+2009-2010+2011 into a table


<table border = "1" cellpadding = "5"><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></table>


Then we'll make a copy of this row and reverse it to place under the current terms
<table border = "1" cellpadding = "5"><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></table>


Add straight down to see each column adds to either 2012 or -2012.
More specifically the odd values add to 2012 while the even values add to -2012.


In the set {1,2,3,...,2010,2011} there are 2010/2 = 1005 even numbers. 
Those even numbers are 2,4,6,...,2010.
This means that we have 1005 instances of the sum -2012 show up.
The other 2011-1005 = 1006 sums are 2012


We then have
1006*2012 + 1005*(-2012)
= 2012*(1006-1005)
= 2012*(1)
= 2012


This is not the final answer. 
It would be nice if 1-2+3-4+...+2009-2010+2011 did evaluate to 2012. 


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


So we have to divide by 2 to correct this error.
2012/2 = <font color=red>1006</font> is the final answer.



In other words, S = 1-2+3-4+...+2009-2010+2011 represents combining terms along the top row
S = 2011-2010+2009-...-4+3-2+1 also happens when we combine terms along the bottom row.
Add those equations straight down to arrive at 2S = 2012 which leads to S = <font color=red>1006</font>
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