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National and Regional Contests
Korea Contests
Korea National Olympiad
2013 Korea National Olympiad
4
4
Part of
2013 Korea National Olympiad
Problems
(1)
Integer sequence with fibonacci formula
Source: Korea National 2013 #4
11/10/2013
{
a
n
}
\{a_n\}
{
a
n
}
is a positive integer sequence such that
a
i
+
2
=
a
i
+
1
+
a
i
(
i
≥
1
)
a_{i+2} = a_{i+1} + a_{i} (i \ge 1)
a
i
+
2
=
a
i
+
1
+
a
i
(
i
≥
1
)
. For positive integer
n
n
n
, define
{
b
n
}
\{b_n\}
{
b
n
}
as
b
n
=
1
a
2
n
+
1
∑
i
=
1
4
n
−
2
a
i
b_n = \frac{1}{a_{2n+1}} \sum_{i=1}^{4n-2} { a_i }
b
n
=
a
2
n
+
1
1
i
=
1
∑
4
n
−
2
a
i
Prove that
b
n
b_n
b
n
is positive integer, and find the general form of
b
n
b_n
b
n
.
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algebra