1996 AMC 8 Problem 20

Question

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

x

currently displayed with the number given by the formula

\frac{1}{1 - x}.

For example, if the calculator is displaying

2

and the special key is pressed, then the calculator will display

-1

since

\frac{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

-0.25

0

0.8

1.25

5

Apply the key a few times starting from

5

and watch for a repeating cycle

The values cycle with period

3;

find where

100

lands in the cycle

Starting from

5:

\frac{1}{1 - 5} = -0.25,

then

\frac{1}{1 + 0.25} = 0.8,

then

\frac{1}{1 - 0.8} = 5.

The values repeat with period

3.

Since

100 = 3 \cdot 33 + 1,

the

100

th press gives the same result as the first press,

-0.25.

Thus, the correct answer is A .

Problem 20 in other years