Difference between revisions of "2004 AMC 10A Problems/Problem 24"
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==Problem== | ==Problem== | ||
− | Let <math>a_1,a_2,\cdots</math>, be a sequence with the following properties. | + | 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>? | What is the value of <math>a_{2^{100}}</math>? | ||
Line 13: | Line 13: | ||
Note that | Note that | ||
− | <math> | + | <math>a_{2^n}=2^{n-1} a_{2^{n -1}}</math> |
− | <math>a_{2^2}=2\cdot | + | 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} = 2^{(1+99)\cdot99/2}=2^{4950}</math> |
− | <math> | + | 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]]. |
− | |||
− | + | ==See also== | |
− | + | {{AMC10 box|year=2004|ab=A|num-b=23|num-a=25}} | |
− | + | [[Category:Introductory Algebra Problems]] |
Revision as of 15:58, 6 February 2007
Problem
Let , be a sequence with the following properties.
(i) , and
(ii) for any positive integer .
What is the value of ?
Solution
Note that
so that
where in the last steps we use the exponent rule and the formula for the sum of an arithmetic series.
See also
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 |