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# Compute the sum

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Compute 1 - 2 + 3 - 4 +... + 2005 - 2006 + 2007 - 2008 + 2009 - 2010.

Nov 4, 2021

#1
+12695
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Compute 1 - 2 + 3 - 4 +... + 2005 - 2006 + 2007 - 2008 + 2009 - 2010.

Hello Guest!

The sum of a finite arithmetic sequence is the number of terms multiplied by the arithmetic mean of the first and the last term.

$$s_n=n\cdot \dfrac{a_1+a_n}{2}$$

n            $$a_1$$       $$a_n$$

$$s_{2009}= \dfrac{1+2009}{2}\cdot \dfrac{1+2009}{2} = 1005\cdot \dfrac{1+2009}{2}=1010025$$

$$s_{-2010}= \dfrac{2+2010}{2}\cdot \dfrac{-2+(-2010)}{2}=1006\cdot \dfrac{-2+(-2010)}{2} =-1012036$$

$$s_n=s_{2009}+s_{-2010}=1010025+(-1012036)$$

$$s_n=-2011$$

!

Nov 5, 2021