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2007 Harvard-MIT Mathematics Tournament
29
29
Part of
2007 Harvard-MIT Mathematics Tournament
Problems
(1)
2007 Guts #29: Infinite Roots of Recursive Sequence
Source:
6/22/2012
A sequence
{
a
n
}
n
≥
1
\{a_n\}_{n\geq 1}
{
a
n
}
n
≥
1
of positive reals is defined by the rule
a
n
+
1
a
n
−
1
5
=
a
n
4
a
n
−
2
2
a_{n+1}a_{n-1}^5=a_n^4a_{n-2}^2
a
n
+
1
a
n
−
1
5
=
a
n
4
a
n
−
2
2
for integers
n
>
2
n>2
n
>
2
together with the initial values
a
1
=
8
a_1=8
a
1
=
8
and
a
2
=
64
a_2=64
a
2
=
64
and
a
3
=
1024
a_3=1024
a
3
=
1024
. Compute
a
1
+
a
2
+
a
3
+
⋯
\sqrt{a_1+\sqrt{a_2+\sqrt{a_3+\cdots}}}
a
1
+
a
2
+
a
3
+
⋯