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Ecuador Mathematical Olympiad (OMEC)
2017 Ecuador NMO (OMEC)
5
5
Part of
2017 Ecuador NMO (OMEC)
Problems
(1)
x_{n + 2} = 3x_{n + 1}-2x_n, y_n = x^2_n+2^{n + 2} 2017 Ecuador NMO (OMEC) 3.5
Source:
9/18/2021
Let the sequences
(
x
n
)
(x_n)
(
x
n
)
and
(
y
n
)
(y_n)
(
y
n
)
be defined by
x
0
=
0
x_0 = 0
x
0
=
0
,
x
1
=
1
x_1 = 1
x
1
=
1
,
x
n
+
2
=
3
x
n
+
1
−
2
x
n
x_{n + 2} = 3x_{n + 1}-2x_n
x
n
+
2
=
3
x
n
+
1
−
2
x
n
for
n
=
0
,
1
,
.
.
.
n = 0, 1, ...
n
=
0
,
1
,
...
and
y
n
=
x
n
2
+
2
n
+
2
y_n = x^2_n+2^{n + 2}
y
n
=
x
n
2
+
2
n
+
2
for
n
=
0
,
1
,
.
.
.
,
n = 0, 1, ...,
n
=
0
,
1
,
...
,
respectively. Show that for all n> 0, and n is the square of a odd integer.
number theory
Perfect Squares
Sequence
Recurrence