2020 AMC 10A 第 8 题

Question

求下列表达式的值：

\begin{aligned} &1+2+3-4+5+6+7-8\\ &\quad+\cdots\\ &\quad+197+198+199-200 \end{aligned}\text{？}

What is the value of

\begin{aligned} &1+2+3-4+5+6+7-8\\ &\quad+\cdots\\ &\quad+197+198+199-200? \end{aligned}

9{,}800

9{,}900

10{,}000

10{,}100

10{,}200

把表达式按每四项一组分组。

Group the expression into blocks of four terms

求第

j

组的公式，再对

50

组求和。

Find a formula for the

j

th block and sum over

50

blocks

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按四项一组：

(1+2+3-4)

+(5+6+7-8)

+\cdots

+(197+198+199-200)

。第

j

组的和为

(4j-3)+(4j-2)

+(4j-1)-4j=8j-6

。

50

组，所以和为

\sum_{j=1}^{50}(8j-6)=8\cdot\dfrac{50\cdot51}{2}

-6\cdot50=9900

。正确答案是 B。

Group the terms in blocks of four:

(1+2+3-4)

+(5+6+7-8)

+\cdots

+(197+198+199-200)

. The

j

th block is

(4j-3)+(4j-2)

+(4j-1)-4j=8j-6

.

There are

50

blocks, so the sum is

\sum_{j=1}^{50}(8j-6)=8\cdot\dfrac{50\cdot51}{2}

-6\cdot50=9900

. Thus, B is the correct answer.

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