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Vietnam National Olympiad
1964 Vietnam National Olympiad
4
4
Part of
1964 Vietnam National Olympiad
Problems
(1)
f_n = (a^{n+1} - b^{n+1})/\sqrt5, where a, b are real such a+b=1, ab=-1, a>b
Source: Vietnamese MO (VMO) 1964
8/22/2018
Define the sequence of positive integers
f
n
f_n
f
n
by
f
0
=
1
,
f
1
=
1
,
f
n
+
2
=
f
n
+
1
+
f
n
f_0 = 1, f_1 = 1, f_{n+2} = f_{n+1} + f_n
f
0
=
1
,
f
1
=
1
,
f
n
+
2
=
f
n
+
1
+
f
n
. Show that
f
n
=
(
a
n
+
1
−
b
n
+
1
)
5
f_n =\frac{ (a^{n+1} - b^{n+1})}{\sqrt5}
f
n
=
5
(
a
n
+
1
−
b
n
+
1
)
, where
a
,
b
a, b
a
,
b
are real numbers such that
a
+
b
=
1
,
a
b
=
−
1
a + b = 1, ab = -1
a
+
b
=
1
,
ab
=
−
1
and
a
>
b
a > b
a
>
b
.
recurrence relation
algebra
Integer sequence