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Algebra Difficulty 3.7 AMC 10/12 Find the answer Canada

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 $f(63)$ ?

Pick one

Solution

We use the functional equation $f(2x+1)=3f(x)$ repeatedly.

Setting $x=1$ , we get $f(3) = 3f(1) = 3 \times 6 = 18$ .

Setting $x=3$ , we get $f(7) = 3f(3) = 3 \times 18 =54$ .

Setting $x=7$ , we get $f(15) = 3f(7) = 3 \times 54 =162$ .

Setting $x=15$ , we get $f(31) = 3f(15) = 3 \times 162 =486$ .

Setting $x=31$ , we get $f(63) = 3f(31) = 3 \times 486 =1458$ .

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.