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MathLinks Contest 5th
3.1
3.1
Part of
MathLinks Contest 5th
Problems
(1)
0531 sequence 5th edition Round 3 p1
Source:
5/6/2021
Let
{
x
n
}
n
\{x_n\}_n
{
x
n
}
n
be a sequence of positive rational numbers, such that
x
1
x_1
x
1
is a positive integer, and for all positive integers
n
n
n
.
x
n
=
2
(
n
−
1
)
n
x
n
−
1
x_n = \frac{2(n - 1)}{n} x_{n-1}
x
n
=
n
2
(
n
−
1
)
x
n
−
1
, if
x
n
1
≤
1
x_{n_1} \le 1
x
n
1
≤
1
x
n
=
(
n
−
1
)
x
n
−
1
−
1
n
x_n = \frac{(n - 1)x_{n-1} - 1}{n}
x
n
=
n
(
n
−
1
)
x
n
−
1
−
1
, if
x
n
1
>
1
x_{n_1} > 1
x
n
1
>
1
. Prove that there exists a constant subsequence of
{
x
n
}
n
\{x_n\}_n
{
x
n
}
n
.
algebra
5th edition