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National and Regional Contests
The Philippines Contests
Philippine MO
2016 Philippine MO
3
Inequality on 1/(i^2 + i)^(3/4)
Inequality on 1/(i^2 + i)^(3/4)
Source: Philippines MO 2016/3
January 21, 2017
inequalities
algebra
Problem Statement
Let
n
n
n
be any positive integer. Prove that
∑
i
=
1
n
1
(
i
2
+
i
)
3
/
4
>
2
−
2
n
+
1
\sum^n_{i=1} \frac{1}{(i^2+i)^{3/4}} > 2-\frac{2}{\sqrt{n+1}}
i
=
1
∑
n
(
i
2
+
i
)
3/4
1
>
2
−
n
+
1
2
.
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