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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=2a_1</math>
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| <math>a_{2^2}=2\cdot a_2=2\cdot1=2</math>
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| <math>a_{2^3}=4\cdot a_4=2^3\cdot2^{2+1}</math>
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| <math>a_{2^8}=8\cdot a_8=2^3\cdot</math>
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| //The following is written by Dale Black -- Not Validated
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| I think it should be <math>2^{4950}</math>
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| <math>a_{2^{100}}=2^{99}\cdot2^{98}\cdot...\cdot2^1\cdot1=2^{(1+99)\cdot99/2}=2^{99\cdot50}=2^{4950}</math>
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