Maximize this!

Algebra Level 5

Let \({ A }_{ 1 },{ A }_{ 2 }, { A }_{ 3 },...,{ A }_{ k }\) be a sequence of positive real numbers. Given that \[\displaystyle \sum _{ i=1 }^{ k }{ { A }_{ i } } = 1000\]

and that \(\prod _{ i=1 }^{ k }{ { A }_{ i } } \) is maximized, determine \(k\).

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