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China Girls Math Olympiad
2016 China Girls Math Olympiad
5
5
Part of
2016 China Girls Math Olympiad
Problems
(1)
Chinese Girls Mathematical Olympiad 2016, Problem 5
Source: China Beijing ,13 Aug 2016
8/14/2016
Define a sequence
{
a
n
}
\{a_n\}
{
a
n
}
by
S
1
=
1
,
S
n
+
1
=
(
2
+
S
n
)
2
4
+
S
n
(
n
=
1
,
2
,
3
,
⋯
)
.
S_1=1,\ S_{n+1}=\frac{(2+S_n)^2}{ 4+S_n} (n=1,\ 2,\ 3,\ \cdots).
S
1
=
1
,
S
n
+
1
=
4
+
S
n
(
2
+
S
n
)
2
(
n
=
1
,
2
,
3
,
⋯
)
.
Where
S
n
S_n
S
n
the sum of first
n
n
n
terms of sequence
{
a
n
}
\{a_n\}
{
a
n
}
. For any positive integer
n
n
n
,prove that
a
n
≥
4
9
n
+
7
.
a_{n}\ge \frac{4}{\sqrt{9n+7}}.
a
n
≥
9
n
+
7
4
.
Sequence
inequalities