AMC 8 1998 Problem 12

AMC8 1998 年第 12 题

数列与规律★★★☆☆

题干整理中

正确答案 · Correct Answer

解析 · Solution

Solution 1

Taking the first product, we have

\displaystyle \left(1 - \frac{1}{2}\right) = \frac{1}{2}

\displaystyle \frac{1}{2} \times 2 = 1

Looking at the second, we get

\displaystyle \left(1 - \frac{1}{3}\right) = \frac{2}{3}

\displaystyle \frac{2}{3} \times 3 = 2

We seem to be going up by 1.

Just to check,

\displaystyle 1 - \frac{1}{n} = \frac{n-1}{n}

\displaystyle \frac{n-1}{n} \times n = n-1

Now that we have discovered the pattern, we have to find the last term.

\displaystyle 1 - \frac{1}{10} = \frac{9}{10}

\displaystyle \frac{9}{10} \times 10 = 9

The sum of all numbers from 1 to 9 is

\displaystyle \frac{9\cdot10}{2} = 45 = A

继续练习