MathDB
Problems
Contests
National and Regional Contests
Romania Contests
District Olympiad
2001 District Olympiad
1
Re: Romania District Olympiad 2001 - Grade X
Re: Romania District Olympiad 2001 - Grade X
Source:
March 16, 2011
induction
arithmetic sequence
algebra proposed
algebra
Problem Statement
Let
(
a
n
)
n
≥
1
(a_n)_{n\ge 1}
(
a
n
)
n
≥
1
be a sequence of real numbers such that
a
1
(
n
1
)
+
a
2
(
n
2
)
+
…
+
a
n
(
n
n
)
=
2
n
−
1
a
n
,
(
∀
)
n
∈
N
∗
a_1\binom{n}{1}+a_2\binom{n}{2}+\ldots+a_n\binom{n}{n}=2^{n-1}a_n,\ (\forall)n\in \mathbb{N}^*
a
1
(
1
n
)
+
a
2
(
2
n
)
+
…
+
a
n
(
n
n
)
=
2
n
−
1
a
n
,
(
∀
)
n
∈
N
∗
Prove that
(
a
n
)
n
≥
1
(a_n)_{n\ge 1}
(
a
n
)
n
≥
1
is an arithmetical progression.Lucian Dragomir
Back to Problems
View on AoPS