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$$n$$ samples of same liquid with masses $$m, 2m, 3m, \ldots, nm$$ at temperatures $$t, 2t, 3t, \ldots, nt$$ are mixed in an isolated system. The resultant temperature of the mixture is
For the same liquid, the final temperature is the mass-weighted mean, $$T = \frac{\sum (km)(kt)}{\sum km} = t\,\frac{\sum k^2}{\sum k}$$. Using $$\sum k^2 = \frac{n(n + 1)(2n + 1)}{6}$$ and $$\sum k = \frac{n(n + 1)}{2}$$, $$T = \frac{(2n + 1)t}{3}$$.
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