MathDB
Problems
Contests
National and Regional Contests
China Contests
ASDAN Math Tournament
2016 ASDAN Math Tournament
10
2016 Algebra #10
2016 Algebra #10
Source:
August 6, 2022
2016
Algebra Test
Problem Statement
Let
a
1
,
a
2
,
…
a_1,a_2,\dots
a
1
,
a
2
,
…
be a sequence of real numbers satisfying
a
n
+
1
a
n
−
a
n
+
2
a
n
−
a
n
+
1
a
n
+
2
a
n
2
=
n
a
n
+
2
a
n
+
1
a
n
.
\frac{a_{n+1}}{a_n}-\frac{a_{n+2}}{a_n}-\frac{a_{n+1}a_{n+2}}{a_n^2}=\frac{na_{n+2}a_{n+1}}{a_n}.
a
n
a
n
+
1
−
a
n
a
n
+
2
−
a
n
2
a
n
+
1
a
n
+
2
=
a
n
n
a
n
+
2
a
n
+
1
.
Given that
a
1
=
−
1
a_1=-1
a
1
=
−
1
and
a
2
=
−
1
2
a_2=-\tfrac{1}{2}
a
2
=
−
2
1
, find the value of
a
9
a
20
\tfrac{a_9}{a_{20}}
a
20
a
9
.
Back to Problems
View on AoPS