Let f be a function with the following properties: (i) f(1) = 1, and (ii) f(2n) = n × f(n) for any positive integer n. What is the value of f(2^100)?

2004 AMC 12A Problem 17 — Solution

2004 AMC 12A Problem 17

Attempt Problem 17 of the 2004 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2004 AMC 12A solutions, or check the answer key.

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17.

Let $f$ be a function with the following properties:

(i) $f(1) = 1,$ and

(ii) $f(2n) = n \cdot f(n)$ for any positive integer $n.$

What is the value of $f(2^{100})?$

$1$

$2^{99}$

$2^{100}$

$2^{4950}$

$2^{9999}$

Answer: D

Concepts:functional equation recursion summation

Difficulty rating: 1720

Solution:

Applying $f(2n) = n \cdot f(n)$ with $n = 2^{k},$ we get $f(2^{k+1}) = 2^{k} \cdot f(2^{k}).$

Unwinding from $f(2^1) = f(2) = 1 \cdot f(1) = 2^0,$ the exponents accumulate: $\begin{aligned} f(2^n) &= 2^{0 + 1 + 2 + \cdots + (n-1)} \\ &= 2^{n(n-1)/2}. \end{aligned}$

Therefore $f(2^{100}) = 2^{100 \cdot 99 / 2} = 2^{4950}.$

Thus, the correct answer is D.

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