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| − | ==Problem==
| + | #REDIRECT [[2004 AMC 12A Problems/Problem 17]] |
| − | Let <math>a_1,a_2,\cdots</math>, be a [[sequence]] with the following properties.
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| − | :(i) <math>a_1=1</math>, and
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| − | :(ii) <math>a_{2n}=n\cdot a_n</math> for any [[positive integer]] <math>n</math>.
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| − | What is the value of <math>a_{2^{100}}</math>?
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| − | <math> \mathrm{(A) \ } 1 \qquad \mathrm{(B) \ } 2^{99} \qquad \mathrm{(C) \ } 2^{100} \qquad \mathrm{(D) \ } 2^{4050} \qquad \mathrm{(E) \ } 2^{9999} </math>
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| − | ==Solution==
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| − | Note that
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| − | <math>a_{2^n}=2^{n-1} a_{2^{n -1}}</math>
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| − | so that <math>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}</math>
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| − | <math>= 2^{(1+99)\cdot99/2}=\boxed{2^{4950}}</math>
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| − | where in the last steps we use the [[exponent]] rule <math>b^x \cdot b^y = b^{x + y}</math> and the formula for the sum of an [[arithmetic series]].
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| − | ==See also==
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| − | {{AMC10 box|year=2004|ab=A|num-b=23|num-a=25}}
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| − | [[Category:Introductory Algebra Problems]]
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