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2011 Kyrgyzstan National Olympiad
8
Sequence problem (KgNM2011)
Sequence problem (KgNM2011)
Source:
April 20, 2011
algebra unsolved
algebra
Problem Statement
Given a sequence
x
1
,
x
2
,
.
.
.
,
x
n
x_1,x_2,...,x_n
x
1
,
x
2
,
...
,
x
n
of real numbers with
x
n
+
1
3
=
x
n
3
−
3
x
n
2
+
3
x
n
{x_{n + 1}}^3 = {x_n}^3 - 3{x_n}^2 + 3{x_n}
x
n
+
1
3
=
x
n
3
−
3
x
n
2
+
3
x
n
, where
(
n
=
1
,
2
,
3
,
.
.
.
)
(n=1,2,3,...)
(
n
=
1
,
2
,
3
,
...
)
. What must be value of
x
1
x_1
x
1
, so that
x
100
x_{100}
x
100
and
x
1000
x_{1000}
x
1000
becomes equal?
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