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National and Regional Contests
PEN Problems
PEN G Problems
24
24
Part of
PEN G Problems
Problems
(1)
G 24
Source:
5/25/2007
Let
{
a
n
}
n
≥
1
\{a_{n}\}_{n \ge 1}
{
a
n
}
n
≥
1
be a sequence of positive numbers such that
a
n
+
1
2
=
a
n
+
1
,
n
∈
N
.
a_{n+1}^{2}= a_{n}+1, \;\; n \in \mathbb{N}.
a
n
+
1
2
=
a
n
+
1
,
n
∈
N
.
Show that the sequence contains an irrational number.
Putnam
Irrational numbers