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South East Mathematical Olympiad
2017 South East Mathematical Olympiad
6
CSMO Grade 10 Problem 6
CSMO Grade 10 Problem 6
Source: China Southeast Mathematical Olympiad
August 1, 2017
algebra
Sequence
Problem Statement
The sequence
{
a
n
}
\{a_n\}
{
a
n
}
satisfies
a
1
=
1
2
a_1 = \frac{1}{2}
a
1
=
2
1
,
a
2
=
3
8
a_2 = \frac{3}{8}
a
2
=
8
3
, and
a
n
+
1
2
+
3
a
n
a
n
+
2
=
2
a
n
+
1
(
a
n
+
a
n
+
2
)
(
n
∈
N
∗
)
a_{n + 1}^2 + 3 a_n a_{n + 2} = 2 a_{n + 1} (a_n + a_{n + 2}) (n \in \mathbb{N^*})
a
n
+
1
2
+
3
a
n
a
n
+
2
=
2
a
n
+
1
(
a
n
+
a
n
+
2
)
(
n
∈
N
∗
)
.
(
1
)
(1)
(
1
)
Determine the general formula of the sequence
{
a
n
}
\{a_n\}
{
a
n
}
;
(
2
)
(2)
(
2
)
Prove that for any positive integer
n
n
n
, there is
0
<
a
n
<
1
2
n
+
1
0 < a_n < \frac{1}{\sqrt{2n + 1}}
0
<
a
n
<
2
n
+
1
1
.
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