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Vietnam National Olympiad
2010 Vietnam National Olympiad
2
VMO 2010-Prove sequence is decreasing
VMO 2010-Prove sequence is decreasing
Source:
December 1, 2010
algebra unsolved
algebra
Problem Statement
Let
{
a
n
}
\{a_{n}\}
{
a
n
}
be a sequence which satisfy
a
1
=
5
a_{1}=5
a
1
=
5
and
a
n
=
a
n
−
1
n
−
1
+
2
n
−
1
+
2.
3
n
−
1
n
∀
n
≥
2
a_{n=}\sqrt[n]{a_{n-1}^{n-1}+2^{n-1}+2.3^{n-1}} \qquad \forall n\geq2
a
n
=
n
a
n
−
1
n
−
1
+
2
n
−
1
+
2.
3
n
−
1
∀
n
≥
2
(a) Find the general fomular for
a
n
a_{n}
a
n
(b) Prove that
{
a
n
}
\{a_{n}\}
{
a
n
}
is decreasing sequences
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