I am struggling generalizing a sequence into a generic formula. The idea is to find a way to represent the account balance for a recurring investment into a dividend stock.
The assumption I am making are: 1) constant return of $p\ \%$ each period; and 2) consistent extra regular investment of $x_{ri}$.
$f(x_0)=x_{ri}$
$f(x_1)=x_{ri} + x_0* (1+p)=x_{ri} + x_{ri} * (1+p)$
$f(x_2)=x_{ri} + x_1*(1+p) = x_{ri} + (x_{ri} + x_{ri} * (1+p))* (1+p)$
$...$
$f(x_{period\ number})\ =\ ... ?$
I have a spreadsheet which calculates the account balance for a set duration of investments, but it would be more convenient to find a single formula to compute the balance.
If I only need to know how the compound interest will affect a one time investment, the formula is as simple as $f(x_{period\ number}) = x_{investment} * (1 + p)^{period\ number}$.
It is the regular investment which make the formula complicated.
You are trying to compute the accumulated value of an annuity. The formula is straightforward to derive if you concentrate instead on the accumulated values of each contribution in the cash flow.
For instance, the initial contribution of $x$ at time $t = 0$ has accumulated value at time $t = n$ of $$x(1+p)^n.$$ The following contribution at time $t = 1$ has accumulated value at time $t = n$ of $$x(1+p)^{n-1}.$$ In general, a contribution of $x$ at time $t = k$ will have accumulated value $$x(1+p)^{n-k}.$$ So the total accumulated value of a series of regular payments will be
$$x(1+p)^n + x(1+p)^{n-1} + \cdots + x(1+p) = x(1+p)\frac{(1+p)^n - 1}{(1+p) - 1} = x(1+p)\frac{(1+p)^n - 1}{p}$$
if we do not count the final payment at time $t = n$ that has no time to accrue interest. This formula corresponds to the accumulated value of a level annuity-due of $x$ per period for $n$ periods at effective interest rate $p$.