2004 AMC 10A Problems/Problem 24

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Problem

Let $a_1,a_2,\cdots$, be a sequence with the following properties.

    (i)  $a_1=1$, and
    (ii)  $a_{2n}=n\cdot a_n$ for any positive integer $n$.

What is the value of $a_{2^{100}}$?

$\mathrm{(A) \ } 1 \qquad \mathrm{(B) \ } 2^{99} \qquad \mathrm{(C) \ } 2^{100} \qquad \mathrm{(D) \ } 2^{4050} \qquad \mathrm{(E) \ } 2^{9999}$

Solution

Note that

$a_2=2a_1$

$a_{2^2}=2\cdot a_2=2\cdot1=2$

$a_{2^3}=4\cdot a_4=2^3\cdot2^{2+1}$

$a_{2^8}=8\cdot a_8=2^3\cdot$

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