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arithmetic mean of $\sqrt[1]{1},\sqrt[2]{2},\sqrt[3]{3},...,\sqrt[n]{n}$ lies in
arithmetic mean of $\sqrt[1]{1},\sqrt[2]{2},\sqrt[3]{3},...,\sqrt[n]{n}$ lies in
Source: SRMC 2012
September 3, 2018
algebra
inequalities
Problem Statement
Prove that for any positive integer
n
n
n
, the arithmetic mean of
1
1
,
2
2
,
3
3
,
…
,
n
n
\sqrt[1]{1},\sqrt[2]{2},\sqrt[3]{3},\ldots ,\sqrt[n]{n}
1
1
,
2
2
,
3
3
,
…
,
n
n
lies in
[
1
,
1
+
2
2
n
]
\left[ 1,1+\frac{2\sqrt{2}}{\sqrt{n}} \right]
[
1
,
1
+
n
2
2
]
.
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