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

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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}

Apply the rule repeatedly with

n = 2^{k}

to unwind

f(2^{100})

The exponent accumulates as

1 + 2 + \cdots + 99

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^{\frac{n(n-1)}{2}}. \end{aligned}

Therefore

f(2^{100}) = 2^{\frac{100 \cdot 99}{2}} = 2^{4950}.

Thus, the correct answer is D.