AMC 8 1985 Problem 2
AMC8 1985 年第 2 题
数列与规律★☆☆☆☆
题干整理中
- A.845
- B.945
- C.1005
- D.1025
- E.1045
正确答案 · Correct Answer
B
解析 · Solution
Solution 1
Rearrange the terms to form 5 pairs, each of which sums to 189: $$$$$$$$
Rewrite this sum as $$$$$$$$
Rewrite this sum as $$$$$$$$
Recall that the sum of an arithmetic series is , where is the number of terms, is the first term and is the last term.
Therefore, the sum of this series is .
Notice that this expression is the sum of the first 99 positive integers minus the sum of the first 89 positive integers, that is, .
Since all the terms are between 90 and 100, the sum must be between and . Only 945 is between 900 and 1000.
Adding all the terms yields 945.
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