2018 AMC 8 Problems/Problem 5

Problem 5

What is the value of $1+3+5+\cdots+2017+2019-2-4-6-\cdots-2016-2018$ ?

$\textbf{(A) }-1010\qquad\textbf{(B) }-1009\qquad\textbf{(C) }1008\qquad\textbf{(D) }1009\qquad \textbf{(E) }1010$

2018 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

Solution

Rearranging the terms, we get $(1-2)+(3-4)+(5-6)+...(2017-2018)+2019$ , and our answer is $-1009+2019=\boxed{1010}, \textbf{(E)}$ - ProMathdunk123

Solution 2 (slightly different)

We can rewrite the given expression as $1+(3-2)+(5-4)+\cdots +(2017-2016)+(2019-2018)=1+1+1+\cdots+1$ . The number of $1$ s is the same as the number of terms in $1,3,5,7\dots ,2017,2019$ . Thus the answer is $\boxed{\textbf{(E) }1010}$

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2018 AMC 8 Problems/Problem 5

Problem 5

Solution

Solution 2 (slightly different)