# 1996 AJHSME Problems/Problem 20

## Problem

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 $\text{(A)}\ -0.25 \qquad \text{(B)}\ 0 \qquad \text{(C)}\ 0.8 \qquad \text{(D)}\ 1.25 \qquad \text{(E)}\ 5$

## Solution

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

After $2$ presses, the calculator displays $\frac{1}{1 - (-\frac{1}{4})} = \frac{1}{\frac{5}{4}} = \frac{4}{5}$

After $3$ presses, the calculator displays $\frac{1}{1 - \frac{4}{5}} = \frac{1}{\frac{1}{5}} = 5$

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 $-\frac{1}{4}$, which is answer $\boxed{A}$.

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 