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PEN Problems
PEN M Problems
29
M 29
M 29
Source:
May 25, 2007
induction
trigonometry
Recursive Sequences
Problem Statement
The sequence
{
a
n
}
n
≥
1
\{a_{n}\}_{n \ge 1}
{
a
n
}
n
≥
1
is defined by
a
1
=
1
a_{1}=1
a
1
=
1
and
a
n
+
1
=
a
n
2
+
1
4
a
n
(
n
∈
N
)
.
a_{n+1}= \frac{a_{n}}{2}+\frac{1}{4a_{n}}\; (n \in \mathbb{N}).
a
n
+
1
=
2
a
n
+
4
a
n
1
(
n
∈
N
)
.
Prove that
2
2
a
n
2
−
1
\sqrt{\frac{2}{2a_{n}^{2}-1}}
2
a
n
2
−
1
2
is a positive integer for
n
>
1
n>1
n
>
1
.
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