Recursion: How can i program this expression?

Question

i have the following expression:

Vt[EA_, V0_, Cm_, B_] := EA + (V0 - Cm)*B

For EA and Cm I have a list of the same length for each. B is a constant. For the first values of the lists the value for V0 is known (3500). For the other values of the lists Mathematica should use the respective outcome of the expression for V0 ( Vt[EA_, V0_, Cm_, B_]=V0 ) and so on. The outcome I want is a list with the same length as the other lists used in the computation +1. So the outcome should be equal to the use of the FoldList[] command. But for the use of the FoldList[] command iI have one expression that is too much. The work with Mathematica is relative new for me so I doen´t know how I can program this kind of recursion.

I hope someone can help me with this problem. Thank you very much

Answer 1

Whenever the current computation depends upon the result of the previous step, think FoldList. FoldList is a fast and efficient function to use for those cases.

Let's look at this for three steps where the inputs are symbolic (tip: don't use upper case symbols, you may end up clashing with system symbols).

vt[ea_, v0_, cm_, b_] := ea + (v0 - cm)*b
ea = {ea1, ea2, ea3};
cm = {cm1, cm2, cm3};

In the application of FoldList we will create a list that consists of {eaN, cmN} pairs.

FoldList[vt[#2[[1]], #1, #2[[2]], b] &, v0,
    Transpose[Join[{ea, cm}]]] // Column

If you study the rows you can see that the v0 input to the current row is the value output from the previous computation.

Now apply it to a numerical example:

v0 = 3500;
ea = {10, 11, 12};
cm = {9, 10, 11};
b = 1.1;

FoldList[vt[#2[[1]], #1, #2[[2]], b] &, v0, 
 Transpose[Join[{ea, cm}]]]

(* {3500, 3850.1, 4235.11, 4658.52} *)

Answer 2

RecurrenceTable[{vt[n + 1] == EA + (vt[n] - Cm)*B, vt[1] == 3500}, vt, {n, 1, 10}]