MathDB
Problems
Contests
National and Regional Contests
Germany Contests
Bundeswettbewerb Mathematik
1994 Bundeswettbewerb Mathematik
2
2
Part of
1994 Bundeswettbewerb Mathematik
Problems
(1)
Bundeswettbewerb Mathematik 1994 Problem 2.2
Source: Bundeswettbewerb Mathematik 1994 Round 2
10/7/2022
Let
k
k
k
be an integer and define a sequence
a
0
,
a
1
,
a
2
,
…
a_0 , a_1 ,a_2 ,\ldots
a
0
,
a
1
,
a
2
,
…
by
a
0
=
0
,
a
1
=
k
and
a
n
+
2
=
k
2
a
n
+
1
−
a
n
for
n
≥
0.
a_0 =0 , \;\; a_1 =k \;\;\text{and} \;\; a_{n+2} =k^{2}a_{n+1}-a_n \; \text{for} \; n\geq 0.
a
0
=
0
,
a
1
=
k
and
a
n
+
2
=
k
2
a
n
+
1
−
a
n
for
n
≥
0.
Prove that
a
n
+
1
a
n
+
1
a_{n+1} a_n +1
a
n
+
1
a
n
+
1
divides
a
n
+
1
2
+
a
n
2
a_{n+1}^{2} +a_{n}^{2}
a
n
+
1
2
+
a
n
2
for all
n
n
n
.
Sequence
number theory
Divisibility