Let a_1, a_2, … be a sequence with the following properties: a_1 = 1, and a_2n = n × a_n for any positive integer n. What is the value of a_2^100?

2004 AMC 10A Problem 24 — Solution

2004 AMC 10A Problem 24

Below is the professionally curated solution for Problem 24 of the 2004 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2004 AMC 10A solutions, or check the answer key.

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Concepts:recursion triangular number pattern recognition

Difficulty rating: 2010

24.

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 $a_{2^{100}}?$

$1$

$2^{99}$

$2^{100}$

$2^{4950}$

$2^{9999}$

Solution:

Applying the rule repeatedly, $a_{2^1} = 2^0, \quad a_{2^2} = 2^1, \quad a_{2^3} = 2^{1+2}, \quad a_{2^4} = 2^{1+2+3}, \ldots$ so in general $a_{2^n} = 2^{1 + 2 + \cdots + (n - 1)} = 2^{n(n-1)/2}.$

For $n = 100,$ the exponent is $\dfrac{100 \cdot 99}{2} = 4950,$ so $a_{2^{100}} = 2^{4950}.$

Thus, the correct answer is D.

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