I have a real world problem that requires me to essentially solve the following equation, for some constant $a$:
$\sum\limits_{j=0}^{n} z_j = E$
where $z_j = z_{j-1} (1 - (1-a)x_j)$, and $z_0 = 1-(1-a)x_0$.
and $E$ is a known constant and the $x_j$ are known for each $j$.
At first I tried to solve the difference equation involving $z_j$ and $z_{j-1}$ but became a little stuck because of the $x_j$ term.
My intention was to solve this difference equation, and then use some form of numerical method e.g. Newton-Raphson to solve for $a$ in the equation involving the desired $E$.
However, I am not sure if this is the 'best' way to approach the problem. I would like to avoid using a 'goal seek' option on the parameter $a$ if possible.
I would appreciate any help or nudge in the right direction, or clarification that there is no 'optimal' way!
Thanks very much