AMC 8 2019 Problem 17
AMC8 2019 年第 17 题
表达式与运算★★★☆☆
What is the value of the product shown below?
下式的值是多少
- A.
- B.
- C.
- D.
- E.50
正确答案 · Correct Answer
B
解析 · Solution
Solution 1
If you calculate the first few values of the equation, all of the values tend to be around , but are not equal to it. The answer closest to but not equal to it is .
Rewriting the numerator and the denominator, we get . We can simplify by canceling 99! on both sides, leaving us with: We rewrite 99! as and cancel 98!, which gets .
All of the terms have the form , which is <1, so the product is <1, so we eliminate options (D) and (E). (C) is too close to 1 to be possible. The partial products seem to be approaching , so we guess that is the limit/asymptote, and so any finite product would be slightly larger than . Therefore, by process of elimination and a small guess, we get that the answer is .
The product can be simplified by observing that in each individual fraction, the numerator and denominator contain factors that cancel out with adjacent terms. Specifically, the factor 3 in the numerator of the first fraction cancels with the 3 in the denominator of the second fraction, the 4 in the numerator of the second fraction cancels with the 4 in the denominator of the third fraction, and so on. This telescoping cancellation continues throughout the entire product. After all the cancellations, only the first factor of the first fraction, , and the last factor of the last fraction, , remain. The value of the product is therefore Thus, the value of the product is .
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