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Algebra Difficulty 3.0 AMC 10/12 Find the answer

Suppose there is a special key on a calculator that replaces the number xx currently displayed with the number given by the formula 1/(1−x)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)=−11/(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

Pick one

Solution

We look for a pattern, hoping this sequence either settles down to one number, or that it forms a cycle that repeats.
After 11 press, the calculator displays 11−5=−14\frac{1}{1 - 5} = -\frac{1}{4}
After 22 presses, the calculator displays 11−(−14)=154=45\frac{1}{1 - (-\frac{1}{4})} = \frac{1}{\frac{5}{4}} = \frac{4}{5}
After 33 presses, the calculator displays 11−45=115=5\frac{1}{1 - \frac{4}{5}} = \frac{1}{\frac{1}{5}} = 5
Thus, every three presses, the display will be 55. On press 3⋅33=993\cdot 33 = 99, the display will be 55. One more press will give −14-\frac{1}{4}, which is answer A\boxed{A}.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.