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PEN M Problems
12
M 12
M 12
Source:
May 25, 2007
modular arithmetic
induction
Recursive Sequences
Problem Statement
Let
k
k
k
be a fixed positive integer. The sequence
{
a
n
}
n
≥
1
\{a_{n}\}_{n\ge1}
{
a
n
}
n
≥
1
is defined by
a
1
=
k
+
1
,
a
n
+
1
=
a
n
2
−
k
a
n
+
k
.
a_{1}=k+1, a_{n+1}=a_{n}^{2}-ka_{n}+k.
a
1
=
k
+
1
,
a
n
+
1
=
a
n
2
−
k
a
n
+
k
.
Show that if
m
≠
n
m \neq n
m
=
n
, then the numbers
a
m
a_{m}
a
m
and
a
n
a_{n}
a
n
are relatively prime.
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