Difference between revisions of "2006 AMC 10B Problems/Problem 18"
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== Problem == | == Problem == | ||
− | Let <math> a_1 , a_2 , ... </math> be a sequence for which | + | Let <math> a_1 , a_2 , ... </math> be a sequence for which <math> a_1=2 </math> , <math> a_2=3 </math>, and <math>a_n=\frac{a_{n-1}}{a_{n-2}} </math> for each positive integer <math> n \ge 3 </math>. What is <math> a_{2006} </math>? |
− | |||
− | <math> a_1=2 </math> , <math> a_2=3 </math>, and <math>a_n=\frac{a_{n-1}}{a_{n-2}} </math> for each positive integer <math> n \ge 3 </math>. | ||
− | |||
− | What is <math> a_{2006} </math>? | ||
<math> \mathrm{(A) \ } \frac{1}{2}\qquad \mathrm{(B) \ } \frac{2}{3}\qquad \mathrm{(C) \ } \frac{3}{2}\qquad \mathrm{(D) \ } 2\qquad \mathrm{(E) \ } 3 </math> | <math> \mathrm{(A) \ } \frac{1}{2}\qquad \mathrm{(B) \ } \frac{2}{3}\qquad \mathrm{(C) \ } \frac{3}{2}\qquad \mathrm{(D) \ } 2\qquad \mathrm{(E) \ } 3 </math> | ||
− | == Solution == | + | == Solution 1 == |
Looking at the first few terms of the sequence: | Looking at the first few terms of the sequence: | ||
Line 17: | Line 13: | ||
Since <math> 2006 \equiv 2\bmod{6}</math>, | Since <math> 2006 \equiv 2\bmod{6}</math>, | ||
− | <math> a_{2006} = a_2 = 3 \ | + | <math> a_{2006} = a_2 = \boxed{\textbf{(E) }3}</math> |
+ | |||
+ | == Solution 2 == | ||
+ | |||
+ | <math> a_n = \frac{a_{n-1}}{a_{n-2}} = \frac{\frac{a_{n-2}}{a_{n-3}}}{a_{n-2}} = \frac{1}{a_{n-3}} </math> , so <math> a_n = a_{n-6} </math> and because <math> 2006 = 2 + 334 \times 6 </math> , so <math> a_{2006} = a_2 = \boxed{\textbf{(E) }3}</math> | ||
+ | |||
+ | ~thatmathsguy | ||
== See Also == | == See Also == | ||
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[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 01:33, 29 May 2023
Contents
Problem
Let be a sequence for which
,
, and
for each positive integer
. What is
?
Solution 1
Looking at the first few terms of the sequence:
Clearly, the sequence repeats every 6 terms.
Since ,
Solution 2
, so
and because
, so
~thatmathsguy
See Also
2006 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
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 |
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