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South Africa Contests
South Africa National Olympiad
2000 South africa National Olympiad
3
Sequence of integers
Sequence of integers
Source: South Africa 2000
September 30, 2005
induction
algebra unsolved
algebra
Problem Statement
Let
c
≥
1
c \geq 1
c
≥
1
be an integer, and define the sequence
a
1
,
a
2
,
a
3
,
…
a_1,\ a_2,\ a_3,\ \dots
a
1
,
a
2
,
a
3
,
…
by
a
1
=
2
,
a
n
+
1
=
c
a
n
+
(
c
2
−
1
)
(
a
n
2
−
4
)
for
n
=
1
,
2
,
3
,
…
.
\begin{aligned} a_1 & = 2, \\ a_{n + 1} & = ca_n + \sqrt{\left(c^2 - 1\right)\left(a_n^2 - 4\right)}\textrm{ for }n = 1,2,3,\dots\ . \end{aligned}
a
1
a
n
+
1
=
2
,
=
c
a
n
+
(
c
2
−
1
)
(
a
n
2
−
4
)
for
n
=
1
,
2
,
3
,
…
.
Prove that
a
n
a_n
a
n
is an integer for all
n
n
n
.
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