AMC10 2020 A

AMC10 2020 A · Q8

AMC10 2020 A · Q8. It mainly tests Algebra misc, Arithmetic sequences basics.

What is the value of $1 + 2 + 3 -4 + 5 + 6 + 7 -8 + \cdots + 197 + 198 + 199 -200$?

求$1 + 2 + 3 -4 + 5 + 6 + 7 -8 + \cdots + 197 + 198 + 199 -200$的值。

(A) 9,800 9,800

(B) 9,900 9,900

(D) 10,100 10,100

(E) 10,200 10,200

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): When terms are combined in groups of four, the sum is the arithmetic series $2+10+18+\cdots+394$ with 50 terms. Its sum is $\dfrac{50}{2}\cdot(2+394)=9,900.$ OR The sum can be viewed as $\displaystyle \sum_{k=1}^{200}k-2\sum_{k=1}^{50}4k=\dfrac{200\cdot201}{2}-8\cdot\dfrac{50\cdot51}{2}=9,900.$

答案（B）：当把各项按四个一组相加时，和成为等差数列 $2+10+18+\cdots+394$，共有 50 项。其和为 $\dfrac{50}{2}\cdot(2+394)=9,900.$ 或者该和也可表示为 $\displaystyle \sum_{k=1}^{200}k-2\sum_{k=1}^{50}4k=\dfrac{200\cdot201}{2}-8\cdot\dfrac{50\cdot51}{2}=9,900.$

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