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National and Regional Contests
Vietnam Contests
Pre - Vietnam Mathematical Olympiad
2011 Pre - Vietnam Mathematical Olympiad
1
Sequence
Sequence
Source: Pre-VMO 2012 - Problem 1
November 27, 2011
induction
algebra proposed
algebra
Problem Statement
Let a sequence
{
x
n
}
\left\{ {{x_n}} \right\}
{
x
n
}
defined by:
{
x
0
=
−
2
x
n
=
1
−
1
−
4
x
n
−
1
2
,
∀
n
≥
1
\left\{ \begin{array}{l} {x_0} = - 2 \\ {x_n} = \frac{{1 - \sqrt {1 - 4{x_{n - 1}}} }}{2},\forall n \ge 1 \\ \end{array} \right.
{
x
0
=
−
2
x
n
=
2
1
−
1
−
4
x
n
−
1
,
∀
n
≥
1
Denote
u
n
=
n
.
x
n
u_n=n.x_n
u
n
=
n
.
x
n
and
v
n
=
∏
i
=
0
n
(
1
+
x
i
2
)
{v_n} = \prod\limits_{i = 0}^n {\left( {1 + x_i^2} \right)}
v
n
=
i
=
0
∏
n
(
1
+
x
i
2
)
. Prove that
{
u
n
}
\left\{ {{u_n}} \right\}
{
u
n
}
,
{
v
n
}
\left\{ {{v_n}} \right\}
{
v
n
}
have finite limit.
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