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Korea Contests
Korea Junior Mathematics Olympiad
2022 Korea Junior Math Olympiad
5
5
Part of
2022 Korea Junior Math Olympiad
Problems
(1)
A strange recurrence relation
Source: kjmo 2022 P5
10/29/2022
A sequence of real numbers
a
1
,
a
2
,
…
a_1, a_2, \ldots
a
1
,
a
2
,
…
satisfies the following conditions.
a
1
=
2
a_1 = 2
a
1
=
2
,
a
2
=
11
a_2 = 11
a
2
=
11
. for all positive integer
n
n
n
,
2
a
n
+
2
=
3
a
n
+
5
(
a
n
2
+
a
n
+
1
2
)
2a_{n+2} =3a_n + \sqrt{5 (a_n^2+a_{n+1}^2)}
2
a
n
+
2
=
3
a
n
+
5
(
a
n
2
+
a
n
+
1
2
)
Prove that
a
n
a_n
a
n
is a rational number for each of positive integer
n
n
n
.
recursion
algebra