Difference between revisions of "2013 AMC 8 Problems/Problem 3"

(See Also)
(thanks to akshaj)
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==Problem==
 
==Problem==
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What is the value of <math>4 \cdot (-1+2-3+4-5+6-7+\cdots+1000)</math>?
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<math>\textbf{(A)}\ -10 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 500 \qquad \textbf{(E)}\ 2000</math>
  
 
==Solution==
 
==Solution==
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Notice that we can pair up every two numbers to make a sum of 1:
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<cmath> \begin{eqnarray*}(-1 + 2 - 3 + 4 - \cdots + 1000) &=& ((-1 + 2) + (-3 + 4) + \cdots + (-999 + 1000)) \ &=& (1 + 1 + \cdots + 1) \  &=& 500</cmath>
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Therefore, the answer is <math>4 \cdot 500= \boxed{\textbf{(E)}\ 2000}</math>.
  
 
==See Also==
 
==See Also==
 
{{AMC8 box|year=2013|num-b=2|num-a=4}}
 
{{AMC8 box|year=2013|num-b=2|num-a=4}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 10:55, 27 November 2013

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 (Error compiling LaTeX. Unknown error_msg)

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

See Also

2013 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 2
Followed by
Problem 4
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

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