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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2020 Vietnam National Olympiad
1
1
Part of
2020 Vietnam National Olympiad
Problems
(1)
VMO-2020
Source: VMO_2020 Day1:P_1
12/27/2019
Let a sequence
(
x
n
)
(x_n)
(
x
n
)
satisfy :
x
1
=
1
x_1=1
x
1
=
1
and
x
n
+
1
=
x
n
+
3
x
n
+
n
x
n
x_{n+1}=x_n+3\sqrt{x_n} + \frac{n}{\sqrt{x_n}}
x
n
+
1
=
x
n
+
3
x
n
+
x
n
n
,
∀
\forall
∀
n
≥
1
\ge1
≥
1
a) Prove lim
n
x
n
=
0
\frac{n}{x_n}=0
x
n
n
=
0
b) Find lim
n
2
x
n
\frac{n^2}{x_n}
x
n
n
2
Sequence