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

$-0.25$

$0$

$0.8$

$1.25$

$5$

Answer: A

Concepts:function modular arithmetic

Difficulty rating: 1280

Small Hint:

Apply the key a few times starting from $5$ and watch for a repeating cycle

Big Hint:

The values cycle with period $3;$ find where $100$ lands in the cycle

Solution:

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 .

1996 AMC 8 Problem 20

Answer: A

Difficulty rating: 1280

Small Hint:

Big Hint:

Solution:

Problem 20 in Other Years