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 $\boxed{A}$ .

1996 AJHSME (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
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1996 AJHSME Problems/Problem 20

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