Question 15
For 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 $n.$
\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 $n.$