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Moldova Contests
Moldova Team Selection Test
1997 Moldova Team Selection Test
6
Prove that $0\leq a_{n}-a_{n+1}<\frac{2}{n^2}, \forall n\in\mathbb{N}.$
Prove that $0\leq a_{n}-a_{n+1}<\frac{2}{n^2}, \forall n\in\mathbb{N}.$
Source: Moldova TST 1997
August 8, 2023
Problem Statement
Let
(
a
n
)
n
∈
N
(a_n)_{n\in\mathbb{N}}
(
a
n
)
n
∈
N
be a sequence of positive numbers such that
a
n
−
2
a
n
+
1
+
a
n
+
2
≥
0
and
∑
j
=
1
n
a
j
≤
1
,
∀
n
∈
N
.
a_n-2a_{n+1}+a_{n+2}\geq 0 \text{ and } \sum_{j=1}^{n} a_j \leq 1, \forall n\in\mathbb{N}.
a
n
−
2
a
n
+
1
+
a
n
+
2
≥
0
and
j
=
1
∑
n
a
j
≤
1
,
∀
n
∈
N
.
Prove that
0
≤
a
n
−
a
n
+
1
<
2
n
2
,
∀
n
∈
N
.
0\leq a_{n}-a_{n+1}<\frac{2}{n^2}, \forall n\in\mathbb{N}.
0
≤
a
n
−
a
n
+
1
<
n
2
2
,
∀
n
∈
N
.
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