2020 AMC 10A Problem 8

What is the value of

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

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Solution

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.