AMC 8 1996 Problem 20

AMC8 1996 年第 20 题

数列与规律★★★☆☆

Suppose there is a special key on a calculator that replaces the number

x

currently displayed with the number given by the formula 1/(1-x) . For example, if the calculator is displaying 2 and the special key is pressed, then the calculator will display -1 since 1/(1-2)=-1 . Now suppose that the calculator is displaying 5. After the special key is pressed 100 times in a row, the calculator will display

假设计算器上有一个特殊键，将当前显示的数

x

替换为公式 1/(1-x) 给出的数。例如，如果计算器显示 2 并按下特殊键，计算器将显示 -1，因为 1/(1-2)=-1。现在假设计算器显示 5。连续按下特殊键 100 次后，计算器将显示

A.-0.25
B.0
C.0.8
D.1.25
E.5

正确答案 · Correct Answer

解析 · Solution

Solution 1

We look for a pattern, hoping this sequence either settles down to one number, or that it forms a cycle that repeats.

After 1 press, the calculator displays

\displaystyle \frac{1}{1 - 5} = -\frac{1}{4}

After 2 presses, the calculator displays

\displaystyle \frac{1}{1 - (-\frac{1}{4})} = \frac{1}{\frac{5}{4}} = \frac{4}{5}

After 3 presses, the calculator displays

\displaystyle \frac{1}{1 - \frac{4}{5}} = \frac{1}{\frac{1}{5}} = 5

Thus, every three presses, the display will be 5. On press

3\cdot 33 = 99

, the display will be 5. One more press will give

\displaystyle -\frac{1}{4}

, which is answer

A

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