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Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2000 China Second Round Olympiad
2
2
Part of
2000 China Second Round Olympiad
Problems
(1)
a_n is always a perfect square.
Source: 2000 China Second Round Olympiad P2
8/13/2019
Define the sequence
a
1
,
a
2
,
…
a_1, a_2, \ldots
a
1
,
a
2
,
…
and
b
1
,
b
2
,
…
b_1, b_2, \ldots
b
1
,
b
2
,
…
as
a
0
=
1
,
a
1
=
4
,
a
2
=
49
a_0=1,a_1=4,a_2=49
a
0
=
1
,
a
1
=
4
,
a
2
=
49
and for
n
≥
0
n \geq 0
n
≥
0
{
a
n
+
1
=
7
a
n
+
6
b
n
−
3
,
b
n
+
1
=
8
a
n
+
7
b
n
−
4.
\begin{cases} a_{n+1}=7a_n+6b_n-3, \\ b_{n+1}=8a_n+7b_n-4. \end{cases}
{
a
n
+
1
=
7
a
n
+
6
b
n
−
3
,
b
n
+
1
=
8
a
n
+
7
b
n
−
4.
Prove that for any non-negative integer
n
,
n,
n
,
a
n
a_n
a
n
is a perfect square.
number theory
Sequence