A number theory problem by Magic Math

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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