2004 AMC 12A Problems/Problem 17

The following problem is from both the 2004 AMC 12A #17 and 2004 AMC 10A #24, so both problems redirect to this page.

Problem

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

$(i)\quad f(1) = 1$, and
$(ii)\quad f(2n) = n\times f(n)$, for any positive integer $n$.

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

$\text {(A)}\ 1 \qquad \text {(B)}\ 2^{99} \qquad \text {(C)}\ 2^{100} \qquad \text {(D)}\ 2^{4950} \qquad \text {(E)}\ 2^{9999}$

Solution

$f(2^{100}) = f(2 \times 2^{99}) = 2^{99} \times f(2^{99})$ $= 2^{99} \cdot 2^{98} \times f(2^{98}) = \ldots$ $= 2^{99}2^{98}\cdots 2^{1} \cdot 1 \cdot f(1)$ $= 2^{99 + 98 + \ldots + 2 + 1}$ $= 2^{\frac{99(100)}{2}} = 2^{4950}$ $\Rightarrow \mathrm{(D)}$.

Video Solution

https://youtu.be/qj5hBxYWalI

See also

2004 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 16
Followed by
Problem 18
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions
2004 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 23
Followed by
Problem 25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 10 Problems and Solutions

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