# The Last Number

Algebra Level 4

$\zeta = 1+4+4+9+9+9+\dots+\underbrace{n^{2}+n^{2}+n^{2}+\dots+n^{2}}_{n \text{ times}}+\dots$

Consider a series of $m$ terms where the number $n^2$ appears $n$ times, with the possible exception of the last number which could be truncated.

If $\zeta$ is the maximum number satisfying the series above, such that $\zeta \leq 250,000$, what is the last number appeared in the series, and for how many times?

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