2004 AMC 10A Problems/Problem 24

Problem

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

    (i)  , and

    (ii)   for any positive integer .

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

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

Note that

$a_2=2a_1$

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

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

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

//The following is written by Dale Black -- Not Validated I think it should be $2^{4950}$

$a_{2^{100}}=2^{99}\cdot2^{98}\cdot...\cdot2^1\cdot1=2^{(1+99)\cdot99/2}=2^{99\cdot50}=2^{4950}$