AMC8 2013

AMC8 2013 · Q3

AMC8 2013 · Q3. It mainly tests Manipulating equations, Arithmetic misc.

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

求 $4 - (-1 + 2 - 3 + 4 - 5 + 6 - 7 + \dots + 1000)$ 的值。

(A) -10 -10

(B) 0 0

(D) 500 500

(E) 2000 2000

Answer

Correct choice: (E)

正确答案：（E）

Solution

We group the addends inside the parentheses two at a time: \begin{align*} -1 + 2 - 3 + 4 - 5 + 6 - 7 + \ldots + 1000 &= (-1 + 2) + (-3 + 4) + (-5 + 6) + \ldots + (-999 + 1000) \\ &= \underbrace{1+1+1+\ldots + 1}_{\text{500 1's}} \\ &= 500. \end{align*} Then the desired answer is $4 \times 500 = \boxed{\textbf{(E)}\ 2000}$.

我们将括号内的加数两两分组： \begin{align*} -1 + 2 - 3 + 4 - 5 + 6 - 7 + \ldots + 1000 &= (-1 + 2) + (-3 + 4) + (-5 + 6) + \ldots + (-999 + 1000) \\ &= \underbrace{1+1+1+\ldots + 1}_{\text{500 1's}} \\ &= 500. \end{align*} 然后所需答案是 $4 \times 500 = \boxed{\textbf{(E)}\ 2000}$。

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