2004 AMC 10A Problem 24

Question

Let

a_1,

a_2,

\ldots

be a sequence with the following properties:

a_1 = 1,

and

a_{2n} = n \cdot a_n

for any positive integer

n.

What is the value of

a_{2^{100}}?

1

2^{99}

2^{100}

2^{4950}

2^{9999}

Compute

a_{2^1},

a_{2^2},

a_{2^3},

a_{2^4}

and track the exponent of

2

The exponents are

0,

1,

1 + 2,

1 + 2 + 3,

\ldots

Applying the rule repeatedly,

\begin{aligned} a_{2^1} &= 2^0, \\ a_{2^2} &= 2^1, \\ a_{2^3} &= 2^{1+2}, \\ a_{2^4} &= 2^{1+2+3}, \ldots \end{aligned}

so in general

a_{2^n} = 2^{1 + 2 + \cdots + (n - 1)} = 2^{\frac{n(n-1)}{2}}.

For

n = 100,

the exponent is

\dfrac{100 \cdot 99}{2} = 4950,

so

a_{2^{100}} = 2^{4950}.

Thus, the correct answer is D.

Problem 24 in other years