Question

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

$-0.25$

$0$

$0.8$

$1.25$

$5$

Answer: A

Concepts:function modular arithmetic

Difficulty rating: 1280

Solution:

Starting from $5$ : $1/(1 - 5) = -0.25$ , then $1/(1 + 0.25) = 0.8$ , then $1/(1 - 0.8) = 5$ . 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 .

1996 AMC 8 Problem 20

Answer: A

Difficulty rating: 1280

Solution:

Problem 20 in Other Years