AMC 8 2018 Problem 5
AMC8 2018 年第 5 题
数列与规律★★☆☆☆
What is the value of the expression shown below?
下面表达式的值是多少
- A.-1010
- B.-1009
- C.1008
- D.1009
- E.1010
正确答案 · Correct Answer
E
解析 · Solution
Solution 1
Rearranging the terms, we get (1-2)+(3-4)+(5-6)+...(2017-2018)+2019, and our answer is -1009+2019=1010. If you were stuck on this problem, refer to AoPS arithmetic lessons.
We can see that the last numbers of each of the sets (even numbers and odd numbers) have a difference of two. So, do the second last ones and so on. Now, all we need to find is the number of integers in any of the sets (I chose even) to get 1010.
It is similar to the: Rearranging the terms, we get 1+(3-2)+(5-4)+(6-5)...(2017-2016)+(2019-2018), and our answer is 1+1009=1010.
Note that the sum of consecutive odd numbers can be expressed as a square, namely . We can modify the negative numbers in the same way by adding 1 to each negative term, factoring a negative sign, and accounting for the extra 1's by subtracting 1009. We then have . Using difference of squares, we obtain (1010+1009)(1010-1009)-1009 = 2019-1009 = 1010.
The sum of odd numbers can be represented as the number of consecutive odds starting from one, squared. Since 2019 is (2019+1)/2, there are 1010 odd numbers. Therefore, the sum of the odd numbers is The sum of even numbers up to 2018 is calculated by noting that there are = 1009 even numbers. Using the formula for the sum of the first n even numbers, n(n+1), we get The difference between these two sums is 1010×1010 - 1009×1010 = 1010. Thus, the correct answer is -1009+2019=1010.
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