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Balkan MO Shortlist
2010 Balkan MO Shortlist
A2
A2
Part of
2010 Balkan MO Shortlist
Problems
(1)
Minimum upper bound on reciprocal terms of a recurrence
Source: Balkan MO ShortList 2010 A2
4/5/2020
Let the sequence
(
a
n
)
n
∈
N
(a_n)_{n \in \mathbb{N}}
(
a
n
)
n
∈
N
, where
N
\mathbb{N}
N
denote the set of natural numbers, is given with
a
1
=
2
a_1=2
a
1
=
2
and
a
n
+
1
a_{n+1}
a
n
+
1
=
=
=
a
n
2
a_n^2
a
n
2
−
-
−
a
n
+
1
a_n+1
a
n
+
1
. Find the minimum real number
L
L
L
, such that for every
k
k
k
∈
\in
∈
N
\mathbb{N}
N
\begin{align*} \sum_{i=1}^k \frac{1}{a_i} < L \end{align*}