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KoMaL A Problems
KoMaL A Problems 2018/2019
A. 738
A. 738
Part of
KoMaL A Problems 2018/2019
Problems
(1)
Sequence defined by quadratic recursion is integral
Source: KöMaL A. 738
2/13/2019
Consider the following sequence:
a
1
=
1
a_1 = 1
a
1
=
1
,
a
2
=
2
a_2 = 2
a
2
=
2
,
a
3
=
3
a_3 = 3
a
3
=
3
, and
a
n
+
3
=
a
n
+
1
2
+
a
n
+
2
2
−
2
a
n
a_{n+3} = \frac{a_{n+1}^2 + a_{n+2}^2 - 2}{a_n}
a
n
+
3
=
a
n
a
n
+
1
2
+
a
n
+
2
2
−
2
for all integers
n
≥
1
n \ge 1
n
≥
1
. Prove that every term of the sequence is a positive integer.
number theory
algebra
recurrence relation