|
|
| (4 intermediate revisions by 4 users not shown) |
| Line 1: |
Line 1: |
| ==Problem==
| | #REDIRECT [[2004 AMC 12A Problems/Problem 17]] |
| Let <math>a_1,a_2,\cdots</math>, be a sequence with the following properties.
| |
| | |
| (i) <math>a_1=1</math>, and
| |
| | |
| (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>?
| |
| | |
| <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>
| |
| | |
| ==Solution==
| |
| Note that
| |
| | |
| <math>a_2=2a_1</math>
| |
| | |
| <math>a_{2^2}=2\cdot a_2=2\cdot1=2</math>
| |
| | |
| <math>a_{2^3}=4\cdot a_4=2^3\cdot2^{2+1}</math>
| |
| | |
| <math>a_{2^8}=8\cdot a_8=2^3\cdot</math>
| |
