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National and Regional Contests
PEN Problems
PEN N Problems
6
N 6
N 6
Source:
May 25, 2007
More Sequences
Problem Statement
Let
{
a
n
}
\{a_{n}\}
{
a
n
}
be a strictly increasing positive integers sequence such that
gcd
(
a
i
,
a
j
)
=
1
\gcd(a_{i}, a_{j})=1
g
cd
(
a
i
,
a
j
)
=
1
and
a
i
+
2
−
a
i
+
1
>
a
i
+
1
−
a
i
a_{i+2}-a_{i+1}>a_{i+1}-a_{i}
a
i
+
2
−
a
i
+
1
>
a
i
+
1
−
a
i
. Show that the infinite series
∑
i
=
1
∞
1
a
i
\sum^{\infty}_{i=1}\frac{1}{a_{i}}
i
=
1
∑
∞
a
i
1
converges.
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