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2015 CentroAmerican
Problem 2
CentroAmerican 2015 #2
CentroAmerican 2015 #2
Source: 2015 CentroAmerican Math Olympiad #2
June 27, 2015
OMCC
algebra
Problem Statement
A sequence
(
a
n
)
(a_n)
(
a
n
)
of real numbers is defined by
a
0
=
1
a_0=1
a
0
=
1
,
a
1
=
2015
a_1=2015
a
1
=
2015
and for all
n
≥
1
n\geq1
n
≥
1
, we have
a
n
+
1
=
n
−
1
n
+
1
a
n
−
n
−
2
n
2
+
n
a
n
−
1
.
a_{n+1}=\frac{n-1}{n+1}a_n-\frac{n-2}{n^2+n}a_{n-1}.
a
n
+
1
=
n
+
1
n
−
1
a
n
−
n
2
+
n
n
−
2
a
n
−
1
.
Calculate the value of
a
1
a
2
−
a
2
a
3
+
a
3
a
4
−
a
4
a
5
+
…
+
a
2013
a
2014
−
a
2014
a
2015
\frac{a_1}{a_2}-\frac{a_2}{a_3}+\frac{a_3}{a_4}-\frac{a_4}{a_5}+\ldots+\frac{a_{2013}}{a_{2014}}-\frac{a_{2014}}{a_{2015}}
a
2
a
1
−
a
3
a
2
+
a
4
a
3
−
a
5
a
4
+
…
+
a
2014
a
2013
−
a
2015
a
2014
.
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