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Baltic Way
2005 Baltic Way
3
Strange sequence
Strange sequence
Source: Baltic Way 2005/3
November 7, 2005
algebra proposed
algebra
Problem Statement
Consider the sequence
{
a
k
}
k
≥
1
\{a_k\}_{k \geq 1}
{
a
k
}
k
≥
1
defined by
a
1
=
1
a_1 = 1
a
1
=
1
,
a
2
=
1
2
a_2 = \frac{1}{2}
a
2
=
2
1
and
a
k
+
2
=
a
k
+
1
2
a
k
+
1
+
1
4
a
k
a
k
+
1
for
k
≥
1.
a_{k + 2} = a_k + \frac{1}{2}a_{k + 1} + \frac{1}{4a_ka_{k + 1}}\ \textrm{for}\ k \geq 1.
a
k
+
2
=
a
k
+
2
1
a
k
+
1
+
4
a
k
a
k
+
1
1
for
k
≥
1.
Prove that
1
a
1
a
3
+
1
a
2
a
4
+
1
a
3
a
5
+
⋯
+
1
a
98
a
100
<
4.
\frac{1}{a_1a_3} + \frac{1}{a_2a_4} + \frac{1}{a_3a_5} + \cdots + \frac{1}{a_{98}a_{100}} < 4.
a
1
a
3
1
+
a
2
a
4
1
+
a
3
a
5
1
+
⋯
+
a
98
a
100
1
<
4.
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