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2013 BMT Spring
17
17
Part of
2013 BMT Spring
Problems
(1)
2013 BMT Individual 17
Source:
1/18/2022
Let
N
≥
1
N \ge 1
N
≥
1
be a positive integer and
k
k
k
be an integer such that
1
≤
k
≤
N
1 \le k \le N
1
≤
k
≤
N
. Define the recurrence
x
n
=
x
n
−
1
+
x
n
−
2
+
.
.
.
+
x
n
−
N
N
x_n = \frac{x_{n-1} + x_{n-2} +... + x_{n-N}}{N}
x
n
=
N
x
n
−
1
+
x
n
−
2
+
...
+
x
n
−
N
for
n
>
N
n > N
n
>
N
and
x
k
=
1
x_k = 1
x
k
=
1
,
x
1
=
x
2
=
.
.
.
=
x
k
−
1
=
x
k
+
1
=
.
.
=
x
N
=
0
x_1 = x_2 = ... = x_{k-1} =x_{k+1} =.. = x_N = 0
x
1
=
x
2
=
...
=
x
k
−
1
=
x
k
+
1
=
..
=
x
N
=
0
. As
n
n
n
approaches infinity,
x
n
x_n
x
n
approaches some value. What is this value?
algebra