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3
Limit of recursive sequence of points - OIMU 2005 Problem 3
Limit of recursive sequence of points - OIMU 2005 Problem 3
Source:
September 3, 2010
limit
algebra proposed
algebra
Problem Statement
Consider the sequence defined recursively by
(
x
1
,
y
1
)
=
(
0
,
0
)
(x_1,y_1)=(0,0)
(
x
1
,
y
1
)
=
(
0
,
0
)
,
(
x
n
+
1
,
y
n
+
1
)
=
(
(
1
−
2
n
)
x
n
−
1
n
y
n
+
4
n
,
(
1
−
1
n
)
y
n
−
1
n
x
n
+
3
n
)
(x_{n+1},y_{n+1})=\left(\left(1-\frac{2}{n}\right)x_n-\frac{1}{n}y_n+\frac{4}{n},\left(1-\frac{1}{n}\right)y_n-\frac{1}{n}x_n+\frac{3}{n}\right)
(
x
n
+
1
,
y
n
+
1
)
=
(
(
1
−
n
2
)
x
n
−
n
1
y
n
+
n
4
,
(
1
−
n
1
)
y
n
−
n
1
x
n
+
n
3
)
.Find
lim
n
→
∞
(
x
n
,
y
n
)
\lim_{n\to \infty}(x_n,y_n)
lim
n
→
∞
(
x
n
,
y
n
)
.
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