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

Latest revision as of 16:11, 5 December 2007

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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.
-     (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>