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MathLinks Contest 4th
3.1
0431 Fibonacci inequality 4th edition Round 3 p1
0431 Fibonacci inequality 4th edition Round 3 p1
Source:
May 7, 2021
algebra
4th edition
Problem Statement
Let
{
f
n
}
n
≥
1
\{f_n\}_{n\ge 1}
{
f
n
}
n
≥
1
be the Fibonacci sequence, defined by
f
1
=
f
2
=
1
f_1 = f_2 = 1
f
1
=
f
2
=
1
, and for all positive integers
n
n
n
,
f
n
+
2
=
f
n
+
1
+
f
n
f_{n+2} = f_{n+1} + f_n
f
n
+
2
=
f
n
+
1
+
f
n
. Prove that the following inequality takes place for all positive integers
n
n
n
:
(
n
1
)
f
1
+
(
n
2
)
f
2
+
.
.
.
+
(
n
n
)
f
n
<
(
2
n
+
2
)
n
n
!
{n \choose 1}f_1 +{n \choose 2}f_2+... +{n \choose n}f_n < \frac{(2n + 2)^n}{n!}
(
1
n
)
f
1
+
(
2
n
)
f
2
+
...
+
(
n
n
)
f
n
<
n
!
(
2
n
+
2
)
n
.
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