Finding an optimal sequence It's my first time on this site:) I have to find a strictly increasing finite sequence $\{x_k\} _{k=1, \dots, n}$ with $x_1=c^2$ that will minimize the following expression $$\sum_{i=1}^n\sqrt{x_{i+1}-x_i} \frac{c}{\sqrt{x_i}},$$ subject to the constraint $x_{n+1}=c^\frac{2}{3}$, where $c$ is a constant less than 1.   
It seems like an optimization problem but in my experience I have only had to minimize expressions with respect to one or few variables. Here, even the number of variables is unknown. I'm having problems understanding how to find an optimal sequence without presupposing its length. Is this even possible and does there exist any thoery on this subject? 
For this particular problem, it makes sense to me to keep adding elements to the sequence in the beginning in order to make use of the $\frac{c}{\sqrt{x_i}}$ factor, and eventually, with enought terms, splitting the interval $[c^2,c^\frac{2}{3}]$ further is harmful because the dominating effect comes from creating a new term in the summation. It also seems the sequence should have more points close to $c^2$ than to the other endpoint, $c^\frac{2}{3}$, but that is as far as I've got.
 A: I think it might be that the lowest sum comes from putting all the intermediate points equal to either $x_1$ or $x_{n+1}$.
As a check:  Fix all $x_j$ except one, $x_i$,  then choose the best value for it.  There are only two terms that involve $x_i$.  Differentiate them with respect to $x_i$, and solve.  The minimum is either where the derivative is zero; or at one end of the interval.  But if it is at one end of the interval, that means either $x_i=x_{i-1}$ or $x_i=x_{i+1}$.  And I think that is what happens.
A: As suggested in another answer, there is no minimum with a strictly increasing sequence. To see this note first that, given that we are dealing with positive numbers only:
$$ \sum_{i=1}^n \sqrt{x_{i+1}-x_i}\frac{c}{\sqrt{x_i}}  = c \sum_{i=1}^n \sqrt{\frac{x_{i+1}}{x_i}-1}$$
For a strictly increasing sequence we can write $x_{i+i}=(1+y_i^2)x_i$ where $y_i^2>0$ is the increment at step $i$. Writing $y_i^2$ rather than $y_i$ is just a matter of convenience, so that the sum simplifies nicely:
$$\sum_{i=1}^n \sqrt{\frac{x_{i+1}}{x_i}-1} = \sum_{i=1}^n \sqrt{1+y_i^2-1} = \sum_{i=1}^n y_i $$
Here we can see easily that the minimum would $y_i=0$ for all $i$ (all $x_i$'s equal to $x_1$, apart from $y_1$ which must be zero per your constraints, and $y_n$ which must be $x_{n+1}-x_1$). However, this choice is forbidden by the constraint that the sequence be increasing.
