2010/11 AHSMC Part 1 Problem 14

Problem 14 of 16

Let $f(x) = x^2$ and $g(x) = x^4$ . We apply $f$ and $g$ alternatively to form $f(x) = x^2,\quad g(f(x)) = g(x^2) = (x^2)^4 = x^8,\quad f(g(f(x))) = f(x^8) = (x^8)^2 = x^{16},$ and so on. After we have applied $f$ $50$ times and $g$ $49$ times, the answer is $x^n$ where $n$ is

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Solution

Each application of $f$ doubles the exponent while each application of $g$ quadruples the exponent. After $50$ applications of $f$ and $49$ applications of $g$ , the exponent has been doubled $50 + 2 \times 49 = 148$ times so that $n = 2^{148}$ . The answer is (C).

University of Alberta

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