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1988 Iran MO (2nd round)
1
The sum of terms is less than 1 - [Iran Second Round 1988]
The sum of terms is less than 1 - [Iran Second Round 1988]
Source:
December 7, 2010
number theory proposed
number theory
Problem Statement
Let
{
a
n
}
n
=
1
∞
\{a_n \}_{n=1}^{\infty}
{
a
n
}
n
=
1
∞
be a sequence such that
a
1
=
1
2
a_1=\frac 12
a
1
=
2
1
and
a
n
=
(
2
n
−
3
2
n
)
a
n
−
1
∀
n
≥
2.
a_n=\biggl( \frac{2n-3}{2n} \biggr) a_{n-1} \qquad \forall n \geq 2.
a
n
=
(
2
n
2
n
−
3
)
a
n
−
1
∀
n
≥
2.
Prove that for every positive integer
n
,
n,
n
,
we have
∑
k
=
1
n
a
k
<
1.
\sum_{k=1}^n a_k <1.
∑
k
=
1
n
a
k
<
1.
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