A number theory problem by Magic Math

Number Theory Level 4

If $$a_1, a_2, \ldots, a_{100}$$ are of the form $$a_i = n + i$$ for some natural number $$n$$, determine the value of $$a_{89}$$ that would make

$\sqrt{ a_2 + a_3 + \ldots + a_{99} } - \sqrt{a_1 + a_{100} }$

the smallest possible positive integer.

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