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National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1988 Swedish Mathematical Competition
6
6
Part of
1988 Swedish Mathematical Competition
Problems
(1)
inequality when a_{n+1} = \sqrt{a_n^2 + 1/a_n }
Source: 1988 Swedish Mathematical Competition p6
3/28/2021
The sequence
(
a
n
)
(a_n)
(
a
n
)
is defined by
a
1
=
1
a_1 = 1
a
1
=
1
and
a
n
+
1
=
a
n
2
+
1
a
n
a_{n+1} = \sqrt{a_n^2 +\frac{1}{a_n}}
a
n
+
1
=
a
n
2
+
a
n
1
for
n
≥
1
n \ge 1
n
≥
1
. Prove that there exists
a
a
a
such that
1
2
≤
a
n
n
a
≤
2
\frac{1}{2} \le \frac{a_n}{n^a} \le 2
2
1
≤
n
a
a
n
≤
2
for
n
≥
1
n \ge 1
n
≥
1
.
inequalities
algebra
Sequence
recurrence relation
radical