Question

What is the sum of the integers from -2007 through +2009, inclusive?

Answer 1

4017

Explanation:

`-2007 + -2006 + -2005 + ... + 2005 + 2006 + 2007 + 2008 + 2009`

By the commutative property of addition, we can rearrange the additives in any order we want and still get the same result

`=> -2007 + 2007 + -2006 + 2006 + -2005 + 2005 + ... + -2 + 2 + -1 + 1 + 0 + 2008 + 2009`

By the associative property of addition, we can change the order of addition, and still get the same result

`=> (-2007 + 2007) + (-2006 + 2006) + (-2005 + 2005) + ... + (-2 + 2) + (-1 + 1) + 0 + 2008 + 2009`

Note that if we add those enclosed in parenthesis, we will get 0,

`=> 0 + 0 + 0 + ... + 0 + 2008 + 2009`

`=> 2008 + 2009`

`=> 4017`

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PS: Please note that we were only able to apply the commutative property and the associative property in such a scale since we are dealing solely with addition. If other operations were involved, we will need to follow PEMDAS

Answer 2

The series formed by integers here is an AP series having

first term `" "a=-2007` , common difference `" "d=+1`

and last term `" "l = 2009`

The total number of terms `" "n=(2009-(-2007)+1=4017`

So Sum`" "S=n/2(a+l)=4017/2(-2007+2009)=4017`