Difference between revisions of "1976 IMO Problems/Problem 6"

Revision as of 10:46, 26 February 2008

A sequence $(u_{n})$ is defined by

$u_{0} = 2 \quad u_{1} = \frac {5}{2}, u_{n + 1} = u_{n}(u_{n - 1}^{2} - 2) - u_{1} \quad \textnormal{for} n = 1,\ldots$

Prove that for any positive integer $n$ we have

$[u_{n}] = 2^{\frac {(2^{n} - ( - 1)^{n})}{3}}$

(where [x] denotes the smallest integer $\leq$ x) $.$

This problem needs a solution. If you have a solution for it, please help us out by adding it.

@@ Line 1: / Line 1: @@
 == Problem ==
-{{problem}}
+A sequence <math>(u_{n})</math> is defined by
+<cmath>u_{0} = 2 \quad u_{1} = \frac {5}{2}, u_{n + 1} = u_{n}(u_{n - 1}^{2} - 2) - u_{1} \quad \textnormal{for} n = 1,\ldots</cmath>
+Prove that for any positive integer <math>n</math> we have
+<cmath>[u_{n}] = 2^{\frac {(2^{n} - ( - 1)^{n})}{3}}</cmath>
+(where [x] denotes the smallest integer <math>\leq</math> x)<math>.</math>
 == Solution ==

1976 IMO (Problems) • Resources
Preceded by Problem 5	1 • 2 • 3 • 4 • 5 • 6	Followed by Final Question
All IMO Problems and Solutions