Problem:
A calculator has a key which replaces the displayed entry with its square, and another key which replaces the displayed entry with its reciprocal. Let be the final result if one starts with an entry and alternately squares and reciprocates times each. Then equals
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Solution:
At each stage the displayed entry can be thought of as a power of . Each reciprocation reverses the sign of the exponent of . Each squaring doubles the exponent. Thus each pair of squaring and reciprocation multiplies the exponent by . Since the initial exponent is , the final exponent is .