2013 AMC 8 Problems/Problem 3

Problem

What is the value of $4 \cdot (-1+2-3+4-5+6-7+\cdots+1000)$ ?

$\textbf{(A)}\ -10 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 500 \qquad \textbf{(E)}\ 2000$

Solution

Notice that we can pair up every two numbers to make a sum of 1: $\begin{eqnarray*}(-1 + 2 - 3 + 4 - \cdots + 1000) &=& ((-1 + 2) + (-3 + 4) + \cdots + (-999 + 1000)) \\ &=& (1 + 1 + \cdots + 1) \\ &=& 500\end{eqnarray*}$

Therefore, the answer is $4 \cdot 500= \boxed{\textbf{(E)}\ 2000}$ .

2013 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
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All AJHSME/AMC 8 Problems and Solutions

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2013 AMC 8 Problems/Problem 3

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