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2000 JBMO ShortLists
12
12
Part of
2000 JBMO ShortLists
Problems
(1)
Two sequences imply a third - JBMO Shortlist
Source:
10/30/2010
Consider a sequence of positive integers
x
n
x_n
x
n
such that:
(
A
)
x
2
n
+
1
=
4
x
n
+
2
n
+
2
(\text{A})\ x_{2n+1}=4x_n+2n+2
(
A
)
x
2
n
+
1
=
4
x
n
+
2
n
+
2
(\text{B})\ x_{3n+\color[rgb]{0.9529,0.0980,0.0118}2}=3x_{n+1}+6x_n for all
n
≥
0
n\ge 0
n
≥
0
. Prove that
(
C
)
x
3
n
−
1
=
x
n
+
2
−
2
x
n
+
1
+
10
x
n
(\text{C})\ x_{3n-1}=x_{n+2}-2x_{n+1}+10x_n
(
C
)
x
3
n
−
1
=
x
n
+
2
−
2
x
n
+
1
+
10
x
n
for all
n
≥
0
n\ge 0
n
≥
0
.
algebra proposed
algebra