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239 Open Math Olympiad
2018 239 Open Mathematical Olympiad
8-9.7
8-9.7
Part of
2018 239 Open Mathematical Olympiad
Problems
(1)
Sequence with strange recurrency
Source: 239 Open MO, 2018, Junior League, Problem 7
4/4/2023
The sequence
a
n
a_n
a
n
is defined by the following conditions:
a
1
=
1
a_1=1
a
1
=
1
, and for any
n
∈
N
n\in \mathbb N
n
∈
N
, the number
a
n
+
1
a_{n+1}
a
n
+
1
is obtained from
a
n
a_n
a
n
by adding three if
n
n
n
is a member of this sequence, and two if it is not. Prove that
a
n
<
(
1
+
2
)
n
a_n<(1+\sqrt 2)n
a
n
<
(
1
+
2
)
n
for all
n
n
n
.Proposed by Mikhail Ivanov
algebra
inequalities