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National and Regional Contests
Romania Contests
District Olympiad
2013 District Olympiad
1
Limit sequence
Limit sequence
Source: Romania District Olympiad 2013,grade XI(Problem 1)
March 10, 2013
limit
inequalities
calculus
calculus computations
Problem Statement
Let
(
a
n
)
n
≥
1
{{\left( {{a}_{n}} \right)}_{n\ge 1}}
(
a
n
)
n
≥
1
an increasing sequence and bounded.Calculate
lim
n
→
∞
(
2
a
n
−
a
1
−
a
2
)
(
2
a
n
−
a
2
−
a
3
)
.
.
.
(
2
a
n
−
a
n
−
2
−
a
n
−
1
)
(
2
a
n
−
a
n
−
1
−
a
1
)
.
\underset{n\to \infty }{\mathop{\lim }}\,\left( 2{{a}_{n}}-{{a}_{1}}-{{a}_{2}} \right)\left( 2{{a}_{n}}-{{a}_{2}}-{{a}_{3}} \right)...\left( 2{{a}_{n}}-{{a}_{n-2}}-{{a}_{n-1}} \right)\left( 2{{a}_{n}}-{{a}_{n-1}}-{{a}_{1}} \right).
n
→
∞
lim
(
2
a
n
−
a
1
−
a
2
)
(
2
a
n
−
a
2
−
a
3
)
...
(
2
a
n
−
a
n
−
2
−
a
n
−
1
)
(
2
a
n
−
a
n
−
1
−
a
1
)
.
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