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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