Difference between revisions of "2007 Cyprus MO/Lyceum/Problem 9"
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==Problem== | ==Problem== | ||
| − | We consider the sequence of real | + | We consider the [[sequence]] of [[real number]]s <math>\displaystyle a_1,a_2,a_3\ldots</math>, such that <math>a_1=0</math>, <math>a_2=1</math> and <math>\displaystyle a_n=a_{n-1}-a_{n-2}</math>, <math>\forall n \in \{3,4,5,6\ldots\}</math>. The value of the term <math>\displaystyle a_{138}</math> is |
<math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } -1\qquad \mathrm{(C) \ } 1\qquad \mathrm{(D) \ } 2\qquad \mathrm{(E) \ } -2</math> | <math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } -1\qquad \mathrm{(C) \ } 1\qquad \mathrm{(D) \ } 2\qquad \mathrm{(E) \ } -2</math> | ||
==Solution== | ==Solution== | ||
| − | The first few terms of the sequence are: | + | The first few terms of the [[recursion|sequence]] are: |
| − | 0, 1, 1, 0, -1, -1, 0, 1, 1, 0 | + | <div style="text-align:center;">0, 1, 1, 0, -1, -1, 0, 1, 1, 0 …</div> |
| − | The sequence repeats every 6 terms | + | The sequence repeats every 6 terms, and <math>\displaystyle 138\equiv0\pmod6</math>. <math>a_{138}=-1\Longrightarrow\mathrm{ B}</math> |
| − | |||
| − | <math>138\equiv0\ | ||
| − | |||
| − | <math>a_{138}=-1\ | ||
==See also== | ==See also== | ||
| − | + | {{CYMO box|year=2007|l=Lyceum|num-b=8|num-a=10}} | |
| − | |||
| − | |||
| − | + | [[Category:Introductory Number Theory Problems]] | |
Latest revision as of 15:39, 6 May 2007
Problem
We consider the sequence of real numbers 
, such that 
, 
and 
, 
. The value of the term 
is
Solution
The first few terms of the sequence are:
0, 1, 1, 0, -1, -1, 0, 1, 1, 0 …
The sequence repeats every 6 terms, and 
. 
See also
| 2007 Cyprus MO, Lyceum (Problems) | ||
| Preceded by Problem 8 |
Followed by Problem 10 | |
| 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 • 26 • 27 • 28 • 29 • 30 | ||
