# A Minimization that is not so Clear

Algebra Level 5

For a positive integer $$n$$, define $$S_n$$ to be the minimum value of the sum $\sum_{k = 1}^n \sqrt{(2k-1)^2 + (a_k)^2},$ where $$a_1, a_2,\ldots, a_n$$ are positive real numbers whose sum is $$17$$. There is a unique positive integer $$n$$ for which $$S_n$$ is also an integer. Find this $$n$$.

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