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Vietnam Contests
Vietnam National Olympiad
1984 Vietnam National Olympiad
2
A sequence satisfying linear recurrence and limit.
A sequence satisfying linear recurrence and limit.
Source: Vietnam MO 1984 P2
March 20, 2011
limit
trigonometry
algebra unsolved
algebra
Problem Statement
The sequence
(
u
n
)
(u_n)
(
u
n
)
is defined by
u
1
=
1
,
u
2
=
2
u_1 = 1, u_2 = 2
u
1
=
1
,
u
2
=
2
and
u
n
+
1
=
3
u
n
−
u
n
−
1
u_{n+1} = 3u_n - u_{n-1}
u
n
+
1
=
3
u
n
−
u
n
−
1
for
n
≥
2
n \ge 2
n
≥
2
. Set
v
n
=
∑
k
=
1
n
arccot
u
k
v_n =\sum_{k=1}^n \text{arccot }u_k
v
n
=
∑
k
=
1
n
arccot
u
k
. Compute
lim
n
→
∞
v
n
\lim_{n\to\infty} v_n
lim
n
→
∞
v
n
.
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