I have been able to classify the recurrence relationship implied by a model I am studying with the following equation:
$E_{n+1}=Xrf+f(r+1)E_n$
where $E_0 = 0$ and thus $E_1 = Xrf$. I would like to know whether a closed-form solution in $X,r,f$ can be written for $E_n$ in terms of elementary functions (summation of finite number of terms is to be expected I suppose), and if it can, then to find that solution. (I am mainly interested in $n>1$)
From evaluating the first few terms, a couple of obvious patterns emerge: $\sum^n_{j=1} Xr^jf^j$ will certainly be a part of the answer, as will $\sum^{n-1}_{j=2}Xrf^j$ (I start from $j=2$ to avoid duplication of the $Xrf$ term). What I am struggling with is the rest - no pattern in the remainder seems clear to me, nor how to generalise to write a summation for it, if that is possible.
Any advice on proceeding with this much appreciated.
This is easier if we temporarily write $Xrf=a,\,f(r+1)=b$ so that the recurrence becomes $E_{n+1}=a+bE_n$. We have $$\begin{align} E_1&=a\\E_2&=a+ab\\E_3&=a+ab+ab^2\\&\vdots\\E_n&=a+ab+ab^2+\cdots+ab^{n-1}\end{align}$$ This last sum is a geometric progression so we have$$E_n={ab^n-a\over b-1}$$ which you can now put back in terms of $X,r,f$ and check that the recurrence is indeed satisfied.