Fermat 2019 Q19

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Problem Statement

The function $f$ has the properties that $f(1) = 6$ and $f(2x + 1) = 3f(x)$ for every integer $x$. What is the value of $f(63)$?

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Solution

$$f(1) = 6$$ $$f(2x + 1) = 3f(x) \;\;\; \forall \; x \in \mathbb{Z}$$ $$f(63) = f(2(31) + 1) = 3f(31)$$ $$f(31) = f(2(15) + 1) = 3f(15)$$ $$f(15) = f(2(7) + 1) = 3f(7)$$ $$f(7) = f(2(3) + 1) = 3f(3)$$ $$f(3) = f(2(1) + 1) = 3f(1)$$ $$\therefore f(63) = 3^5f(1) = 3^5(6) = 1458$$