2008 iTest Problems/Problem 41
Problem
Suppose that 
.
Find the value of 
, where 
.
Solution
Note that for a given integer 
, where 
,
![\[x_a + a = \sum_{n=1}^{2008}x_n + 2009\]](http://latex.artofproblemsolving.com/2/6/0/26058e4a056d4e062addb4bf123f16423b07e350.png)
Add up the equations for all to get
![\[\sum_{n=1}^{2008}x_n + \frac{2009 \cdot 2008}{2} = 2008(\sum_{n=1}^{2008}x_n + 2009)\]](http://latex.artofproblemsolving.com/5/c/9/5c9844d0246f6e9c17049123c16e04ae556356d4.png)
We can substitue and simplify to make the equation look easier to solve.
![\[S + 2009 \cdot 1004 = 2008S + 2009 \cdot 2008\]](http://latex.artofproblemsolving.com/6/c/b/6cbf5a89e96a80c9a21e2dd244acf231b308c635.png)
![\[-2007S = 2009 \cdot 1004\]](http://latex.artofproblemsolving.com/e/6/a/e6aa2a8f3040b42a61fbb41aa27ec42fee92fe94.png)
![\[S = \frac{2009 \cdot 1004}{-2007}\]](http://latex.artofproblemsolving.com/0/8/e/08ead36c077b2dea8fa91a719b0d5f194201a978.png)
Thus, 
.
See Also
| 2008 iTest (Problems) | ||
| Preceded by: Problem 40 |
Followed by: Problem 42 | |
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