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Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2011 China Second Round Olympiad
10
sequence a_n
sequence a_n
Source: China Second Round Test1
January 30, 2012
integration
algebra unsolved
algebra
Problem Statement
A sequence
a
n
a_n
a
n
satisfies
a
1
=
2
t
−
3
a_1 =2t-3
a
1
=
2
t
−
3
(
t
≠
1
,
−
1
t \ne 1,-1
t
=
1
,
−
1
), and
a
n
+
1
=
(
2
t
n
+
1
−
3
)
a
n
+
2
(
t
−
1
)
t
n
−
1
a
n
+
2
t
n
−
1
a_{n+1}=\dfrac{(2t^{n+1}-3)a_n+2(t-1)t^n-1}{a_n+2t^n-1}
a
n
+
1
=
a
n
+
2
t
n
−
1
(
2
t
n
+
1
−
3
)
a
n
+
2
(
t
−
1
)
t
n
−
1
.i) Find
a
n
a_n
a
n
,ii) If
t
>
0
t>0
t
>
0
, compare
a
n
+
1
a_{n+1}
a
n
+
1
with
a
n
a_n
a
n
.
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