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Korea Contests
Korea National Olympiad
2013 Korea National Olympiad
7
Integer sequences
Integer sequences
Source: Korea National 2013 #7
November 10, 2013
Vieta
algebra
Problem Statement
For positive integer
k
k
k
, define integer sequence
{
b
n
}
,
{
c
n
}
\{ b_n \}, \{ c_n \}
{
b
n
}
,
{
c
n
}
as follows:
b
1
=
c
1
=
1
b_1 = c_1 = 1
b
1
=
c
1
=
1
b
2
n
=
k
b
2
n
−
1
+
(
k
−
1
)
c
2
n
−
1
,
c
2
n
=
b
2
n
−
1
+
c
2
n
−
1
b_{2n} = kb_{2n-1} + (k-1)c_{2n-1}, c_{2n} = b_{2n-1} + c_{2n-1}
b
2
n
=
k
b
2
n
−
1
+
(
k
−
1
)
c
2
n
−
1
,
c
2
n
=
b
2
n
−
1
+
c
2
n
−
1
b
2
n
+
1
=
b
2
n
+
(
k
−
1
)
c
2
n
,
c
2
n
+
1
=
b
2
n
+
k
c
2
n
b_{2n+1} = b_{2n} + (k-1)c_{2n}, c_{2n+1} = b_{2n} + kc_{2n}
b
2
n
+
1
=
b
2
n
+
(
k
−
1
)
c
2
n
,
c
2
n
+
1
=
b
2
n
+
k
c
2
n
Let
a
k
=
b
2014
a_k = b_{2014}
a
k
=
b
2014
. Find the value of
∑
k
=
1
100
(
a
k
−
a
k
2
−
1
)
1
2014
\sum_{k=1}^{100} { (a_k - \sqrt{{a_k}^2-1} )^{ \frac{1}{2014}} }
k
=
1
∑
100
(
a
k
−
a
k
2
−
1
)
2014
1
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