MathDB
Problems
Contests
National and Regional Contests
The Philippines Contests
Philippine MO
2009 Philippine MO
1
1
Part of
2009 Philippine MO
Problems
(1)
2009 Philippines MO P1
Source:
11/16/2015
The sequence
a
0
,
a
1
,
a
2
,
.
.
.
{a_0, a_1, a_2, ...}
a
0
,
a
1
,
a
2
,
...
of real numbers satisfies the recursive relation
n
(
n
+
1
)
a
n
+
1
+
(
n
−
2
)
a
n
−
1
=
n
(
n
−
1
)
a
n
n(n+1)a_{n+1}+(n-2)a_{n-1} = n(n-1)a_n
n
(
n
+
1
)
a
n
+
1
+
(
n
−
2
)
a
n
−
1
=
n
(
n
−
1
)
a
n
for every positive integer
n
n
n
, where
a
0
=
a
1
=
1
a_0 = a_1 = 1
a
0
=
a
1
=
1
. Calculate the sum
a
0
a
1
+
a
1
a
2
+
.
.
.
+
a
2008
a
2009
\frac{a_0}{a_1} + \frac{a_1}{a_2} + ... + \frac{a_{2008}}{a_{2009}}
a
1
a
0
+
a
2
a
1
+
...
+
a
2009
a
2008
.
algebra
Sequences
PMO
2009