Background--Simple iteration from a Fixed Boundary Condition

Suppose a function,  F[i], changes proportional to its current size, i.e.,  F[i+1] = F[i] + αF[i]

In the above, the symbol is assigned (ExplFun[i,alpha] = ...) as part of the function definition, so that intermediate values are ``remembered.''
The function needs some value at some time (an initial condition) from which it obtains all its other values:

Created by Wolfram Mathematica 6.0  (06 November 2007)