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Serbia Contests
Serbia Team Selection Test
1996 Yugoslav Team Selection Test
Problem 3
Problem 3
Part of
1996 Yugoslav Team Selection Test
Problems
(1)
sequence is sum of consecutive squares
Source: Yugoslav TST 1996 P3
5/16/2021
The sequence
{
x
n
}
\{x_n\}
{
x
n
}
is given by
x
n
=
1
4
(
(
2
+
3
)
2
n
−
1
+
(
2
−
3
)
2
n
−
1
)
,
n
∈
N
.
x_n=\frac14\left(\left(2+\sqrt3\right)^{2n-1}+\left(2-\sqrt3\right)^{2n-1}\right),\qquad n\in\mathbb N.
x
n
=
4
1
(
(
2
+
3
)
2
n
−
1
+
(
2
−
3
)
2
n
−
1
)
,
n
∈
N
.
Prove that each
x
n
x_n
x
n
is equal to the sum of squares of two consecutive integers.
number theory
Sequences