1996 AJHSME Problems/Problem 20

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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 $\boxed{A}$.

See Also

1996 AJHSME (ProblemsAnswer KeyResources)
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
All AJHSME/AMC 8 Problems and Solutions

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