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

Revision as of 16:31, 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.

2004 AMC 10A (Problems • Answer Key • Resources)
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

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

Page

Toolbox

Search

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

Revision as of 16:31, 6 February 2007

Problem

Solution

See also