The Last Number

Algebra Level 4

ζ=1+4+4+9+9+9++n2+n2+n2++n2n times+\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 m terms where the number n2 n^2 appears n n times, with the possible exception of the last number which could be truncated.

If ζ \zeta is the maximum number satisfying the series above, such that ζ250,000\zeta \leq 250,000, what is the last number appeared in the series, and for how many times?

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