Why exactly for this minima condition is: path difference = $(2n-1)\lambda/2,$ instead of $(2n+1)\lambda/2$?
And what is the starting order of minimal here?
Also, what does a negative path difference mean?
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Let $m=n-1$. Then $(2n-1) = (2m+1)$. Therefore your two expressions are exactly the same if all integer values of $n$ are allowed; it is just a question of where you start the count.
If the path difference is $x_2 - x_1$ for two paths of lengths $x_1,\;x_2$ then a negative path difference means simply that $x_1 > x_2$. It means the first route was longer than the second, rather than shorter than the second.