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National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2013 Korea Junior Math Olympiad
3
3
Part of
2013 Korea Junior Math Olympiad
Problems
(1)
b_n=\frac{1}{a_{2n+1}}\Sigma_{i=1}^{4n-2}a_i is positive integer
Source: KJMO 2013 p3
5/1/2019
{
a
n
}
\{a_n\}
{
a
n
}
is a positive integer sequence such that
a
i
+
2
=
a
i
+
1
+
a
i
a_{i+2} = a_{i+1} +a_i
a
i
+
2
=
a
i
+
1
+
a
i
(for all
i
≥
1
i \ge 1
i
≥
1
). For positive integer
n
n
n
, de fine as
b
n
=
1
a
2
n
+
1
Σ
i
=
1
4
n
−
2
a
i
b_n=\frac{1}{a_{2n+1}}\Sigma_{i=1}^{4n-2}a_i
b
n
=
a
2
n
+
1
1
Σ
i
=
1
4
n
−
2
a
i
Prove that
b
n
b_n
b
n
is positive integer.
algebra
Sequence
recurrence relation
Sum
positive integer
Diophantine equation