We have a bag of tokens. Each token has a label $m$ chosen from the squares of the first $n$ natural numbers, with frequency $\sqrt{m}$. What is the arithmetic mean of the labels?

The tokens have labels from the squares of the first $n$ natural numbers: $1^2, 2^2, 3^2, \ldots, n^2$.

Each label $m$ appears with frequency $\sqrt{m}$:

• Label $1^2 = 1$ appears $\sqrt{1} = 1$ time
• Label $2^2 = 4$ appears $\sqrt{4} = 2$ times
• Label $3^2 = 9$ appears $\sqrt{9} = 3$ times
• Label $k^2$ appears $\sqrt{k^2} = k$ times
• Label $n^2$ appears $\sqrt{n^2} = n$ times

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The total number of tokens is:

$\text{Total tokens} = 1 + 2 + 3 + \cdots + n$

$= \sum_{k=1}^{n} k$

$= \frac{n(n+1)}{2}$

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The sum of all label values is:

$\text{Sum} = 1^2 \cdot 1 + 2^2 \cdot 2 + 3^2 \cdot 3 + \cdots + n^2 \cdot n$

$= \sum_{k=1}^{n} k^2 \cdot k$

$= \sum_{k=1}^{n} k^3$

$= \left(\frac{n(n+1)}{2}\right)^2$

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The arithmetic mean of the labels is:

$\text{Mean} = \frac{\text{Sum of all label values}}{\text{Total number of tokens}}$

$= \frac{\left(\frac{n(n+1)}{2}\right)^2}{\frac{n(n+1)}{2}}$

$= \frac{n(n+1)}{2}$

Therefore, the arithmetic mean of the labels is $\frac{n(n+1)}{2}$.