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Problems
Contests
National and Regional Contests
Latvia Contests
Latvia TST
2020 Latvia TST
1.2
1.2
Part of
2020 Latvia TST
Problems
(1)
Sequence
Source: 2020 Latvia TST Round 1
3/30/2020
Sequences
(
a
n
)
,
(
b
n
)
(a_n), (b_n)
(
a
n
)
,
(
b
n
)
are defined by
a
1
=
1
,
b
1
=
2
a_1 = 1, b_1 = 2
a
1
=
1
,
b
1
=
2
,
a
n
+
1
=
1
+
a
n
+
a
n
b
n
b
n
a_{n+1} = \frac{ 1 + a_n + a_nb_n}{b_n}
a
n
+
1
=
b
n
1
+
a
n
+
a
n
b
n
,
b
n
+
1
=
1
+
b
n
+
a
n
b
n
a
n
b_{n+1} = \frac{ 1 +b_n+ a_nb_n}{a_n}
b
n
+
1
=
a
n
1
+
b
n
+
a
n
b
n
for all positive integers
n
n
n
. Prove that
a
2020
<
5
a_{2020} < 5
a
2020
<
5
.
algebra