MathDB
Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
2006 China Team Selection Test
1
Find general term
Find general term
Source: China TST Quiz 2006
June 18, 2006
induction
ratio
inequalities
limit
algebra unsolved
algebra
Problem Statement
Two positive valued sequences
{
a
n
}
\{ a_{n}\}
{
a
n
}
and
{
b
n
}
\{ b_{n}\}
{
b
n
}
satisfy: (a):
a
0
=
1
≥
a
1
a_{0}=1 \geq a_{1}
a
0
=
1
≥
a
1
,
a
n
(
b
n
+
1
+
b
n
−
1
)
=
a
n
−
1
b
n
−
1
+
a
n
+
1
b
n
+
1
a_{n}(b_{n+1}+b_{n-1})=a_{n-1}b_{n-1}+a_{n+1}b_{n+1}
a
n
(
b
n
+
1
+
b
n
−
1
)
=
a
n
−
1
b
n
−
1
+
a
n
+
1
b
n
+
1
,
n
≥
1
n \geq 1
n
≥
1
. (b):
∑
i
=
1
n
b
i
≤
n
3
2
\sum_{i=1}^{n}b_{i}\leq n^{\frac{3}{2}}
∑
i
=
1
n
b
i
≤
n
2
3
,
n
≥
1
n \geq 1
n
≥
1
. Find the general term of
{
a
n
}
\{ a_{n}\}
{
a
n
}
.
Back to Problems
View on AoPS