AMC 8 1996 Problem 4
AMC8 1996 年第 4 题
数列与规律★★☆☆☆
题干整理中
- A.
- B.
- C.
- D.
- E.
正确答案 · Correct Answer
B
解析 · Solution
Solution 1
First, notice that each number in the numerator is a multiple of 2, and each number in the denominator is a multiple of 3. This suggests that each expression can be factored. Factoring gives:
Since all the material in the parentheses is the same, the common factor in the numerator and the denominator may be cancelled, leaving , which is option .
There are 17 terms in the numerator 2 + 4 + 6 +... + 30 + 32+ 34. Consider adding those terms, but in a different order. Start with the last two terms, 2 + 34, and then add the next two terms on the outside, 4 + 32, and continue. You will get 8 pairs of numbers that add to 36, while the number in the middle will be alone. That number is 18. Adding all the numbers gives .
Similarly, the denominator has 17 terms of 3 + 6 + 9 +... + 45 + 48 + 51. There are 8 pairs of numbers that add up to 54, with the number in the center being 27. The total of all the numbers is .
The answer is . Eyeballing the options, the fraction is clearly under 1, but more than . Thus, the answer must be , or . Alternately, you can do the work by factoring out a 9 in the numerator to give . Factoring out 17 will give the desired answer.
Start by finding a pattern: , , . Each step doesn't seem to change the value of the fraction, so is the right answer.
继续练习