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Tuymaada Olympiad
2016 Tuymaada Olympiad
1
Easy sequence
Easy sequence
Source: tuymaada 2016, P1 Seniors
July 22, 2016
algebra
Sequence
easy
Olympiad
algebra solved
Problem Statement
The sequence
(
a
n
)
(a_n)
(
a
n
)
is defined by
a
1
=
0
a_1=0
a
1
=
0
,
a
n
+
1
=
a
1
+
a
2
+
…
+
a
n
n
+
1.
a_{n+1}={a_1+a_2+\ldots+a_n\over n}+1.
a
n
+
1
=
n
a
1
+
a
2
+
…
+
a
n
+
1.
Prove that
a
2016
>
1
2
+
a
1000
a_{2016}>{1\over 2}+a_{1000}
a
2016
>
2
1
+
a
1000
.
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