Difference between revisions of "2004 AMC 10A Problems/Problem 24"

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Let <math>a_1,a_2,\cdots</math>, be a [[sequence]] with the following properties.
 
Let <math>a_1,a_2,\cdots</math>, be a [[sequence]] with the following properties.
  
    (i)  <math>a_1=1</math>, and
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:(i)  <math>a_1=1</math>, and
  
    (ii)  <math>a_{2n}=n\cdot a_n</math> for any [[positive integer]] <math>n</math>.
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:(ii)  <math>a_{2n}=n\cdot a_n</math> for any [[positive integer]] <math>n</math>.
  
 
What is the value of <math>a_{2^{100}}</math>?
 
What is the value of <math>a_{2^{100}}</math>?

Revision as of 17:31, 6 February 2007

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

$a_{2^n}=2^{n-1} a_{2^{n -1}}$

so that $a_{2^{100}} = 2^{99}\cdot a_{2^{99}} = 2^{99} \cdot 2^{98} \cdot a_{2^{98}} = \cdots = 2^{99}\cdot2^{98}\cdot\cdots\cdot2^1\cdot2^0 \cdot a_{2^0} = 2^{(1+99)\cdot99/2}=2^{4950}$

where in the last steps we use the exponent rule $b^x \cdot b^y = b^{x + y}$ and the formula for the sum of an arithmetic series.


See also

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