MathDB
Problems
Contests
National and Regional Contests
China Contests
South East Mathematical Olympiad
2021 South East Mathematical Olympiad
1
1
Part of
2021 South East Mathematical Olympiad
Problems
(1)
China South East Mathematical Olympiad 2021 Grade10 P1
Source:
7/28/2021
A sequence
{
a
n
}
\{a_n\}
{
a
n
}
is defined recursively by
a
1
=
1
2
,
a_1=\frac{1}{2},
a
1
=
2
1
,
and for
n
≥
2
,
n\ge 2,
n
≥
2
,
0
<
a
n
≤
a
n
−
1
0<a_n\leq a_{n-1}
0
<
a
n
≤
a
n
−
1
and
a
n
2
(
a
n
−
1
+
1
)
+
a
n
−
1
2
(
a
n
+
1
)
−
2
a
n
a
n
−
1
(
a
n
a
n
−
1
+
a
n
+
1
)
=
0.
a_n^2(a_{n-1}+1)+a_{n-1}^2(a_n+1)-2a_na_{n-1}(a_na_{n-1}+a_n+1)=0.
a
n
2
(
a
n
−
1
+
1
)
+
a
n
−
1
2
(
a
n
+
1
)
−
2
a
n
a
n
−
1
(
a
n
a
n
−
1
+
a
n
+
1
)
=
0.
(
1
)
(1)
(
1
)
Determine the general formula of the sequence
{
a
n
}
;
\{a_n\};
{
a
n
}
;
(
2
)
(2)
(
2
)
Let
S
n
=
a
1
+
⋯
+
a
n
.
S_n=a_1+\cdots+a_n.
S
n
=
a
1
+
⋯
+
a
n
.
Prove that for
n
≥
1
,
n\ge 1,
n
≥
1
,
ln
(
n
2
+
1
)
<
S
n
<
ln
(
n
+
1
)
.
\ln\left(\frac{n}{2}+1\right)<S_n<\ln(n+1).
ln
(
2
n
+
1
)
<
S
n
<
ln
(
n
+
1
)
.
Sequence
algebra