Question

求下列表达式的值： $\begin{aligned} &1+2+3-4+5+6+7-8\\ &\quad+\cdots\\ &\quad+197+198+199-200? \end{aligned}$

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

$9,800$

$9,900$

$10,000$

$10,100$

$10,200$

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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 $(4j-3)+(4j-2)$ $+(4j-1)-4j=8j-6$ 。

共有 $50$ 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.