AMC 8 2013 Problem 17
AMC8 2013 年第 17 题
数列与规律★★★☆☆
The sum of six consecutive positive integers is 2013. What is the largest of these six integers?
六个连续正整数之和是 2013，那么这六个整数中最大的整数是多少？
- A.335
- B.338
- C.340
- D.345
- E.350
正确答案 · Correct Answer
B
解析 · Solution
Solution 1
The arithmetic mean of these numbers is . Therefore the numbers are 333, 334, 335, 336, 337, 338, so the answer is 338.
Let the number be . Then our desired number is x+2. Our integers are x-3,x-2,x-1,x,x+1,x+2, so we have that .
Let the first term be . Our integers are x,x+1,x+2,x+3,x+4,x+5. We have, .
Since there are 6 numbers, we divide 2013 by 6 to find the mean of the numbers. . Then, (the fourth number). Fifth: 337; Sixth: 338.
Let the 6th number be . Then our list is: x-6+x-5+x-4+x-3+x-x-1=2013. Simplifying this gets you , which means that x = 338.
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