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Question

A calculator has a key that replaces the displayed entry with its square, and another key which replaces the displayed entry with its reciprocal. Let y be the final result when one starts with a number x=0 and alternately squares and reciprocates n times each. Assuming the calculator is completely accurate (e.g. no roundoff or overflow), then y equals

Accepted Answer

Notice that taking the reciprocal of a number is equivalent to raising the number to the -1st power. Therefore, taking the reciprocal of a number x and then squaring it is (x^-1)^2=x^-2. Also, since multiplication is commutative, this is equivalent to squaring the number then taking the reciprocal of it. Doing this n times equals ((x^-2)^-2)^)) (n times)=x^(-2)^n, x^((-2)^n).