$\displaystyle \sum_{i=1}^{n-1} \sqrt{(a+b)^2}$
a and b are just random constants. I would like to minimize the sum the way they explain it at the bottom of this page:
Sum -- from Wolfram MathWorld
What you have shown is simply $\displaystyle (n- 1)|a+b|$. You can minimize that by taking, for example, any value for a and then taking b= -a.
Surely that's NOT what you intended to say.
Notice that in the example you link to, the sum is over the index i and there is a $\displaystyle x_i$ in the summand so you have different terms for different i in your sum. Your example does not have that. I don't think anyone can answer what you meant to ask until you clarify what you are asking.