I am wondering how one can solve generic 'summations' (or actually: if the idea I had to use integrals is correct):
To bring it to the point I have the following sum equation:
$$\sum_{i=0}^{19} \frac{1}{\epsilon}(b-ax_i) = 5$$ $$x_i = L\cdot (i + \frac{1}{2})$$ Here $b$ and $a$ are knowns, and I have to solve it and find the correct $\epsilon$ - and $x_i$ is simply the middle of a certain segment (each segment has the same length). The idea I had was just to find the integral to $x$, and change the limits such that the max is simply $x_{19} (=1.825\cdot19.5)$.
So: $$\int_0^{19.5L} \frac{1}{\epsilon}(b-ax) dx =5$$ $$b\cdot19.5L + a \cdot 19.5^2L^2 = 5 \epsilon$$ $$\epsilon = 5.1L \cdot (b+a\cdot 19.5 L)$$
Edit, same problem, but even more "pressing/important" (notice especially the $i-1$ term): $$\sum_0^{19}\delta_i + 2 \delta_{i-1}$$ $$\delta_i = a\cdot x_i = a \cdot L \cdot i$$ $$\delta_{i<0} = 0$$