What's the equation for savings growth, that considers contributions?

Question

I'm looking for the equation that basically works like this

"Assuming you have X saved, and save Y more a year, and your savings grow at Z percent, you will have S after T years."

Answer 1

If

• $X is the amount saved as of the beginning of the time period,

• $Y is the amount contributed at fixed intervals of time (once a month, once a quarter, once a year, whatever), and the contribution is made at the end of the interval

and

• during each such interval, the earnings on the account are 100z% for that time period,

then,

• at the end of the first interval, the account will have $X(1+z)+Y in it

• at the end of the second interval, the account will have $(X(1+z)+Y)(1+z)+Y = X(1+z)^2 + Y(1+z) + Y in it

• at the end of the third interval, the account will have
  $(X(1+z)^2+Y(1+z)+Y)(1+z)+Y = X(1+z)^3 + Y(1+z)^2 + Y(1+z) + Y in it

money.SE does not support MathJax and so additional math gets messier, but some people may have recognized the calculations as being the same as what is called Horner's rule or Horner's method for evaluating a polynomial. The polynomial in question is a polynomial with variable t given by

Xt^n + Yt^{n-1} + Yt^{n-2} + .... + Yt + Y

and it is evaluated at t = 1+z. Note that if the periodic contributions are variable instead of being fixed, this is easily accommodated by changing the appropriate coefficient of the polynomial. On the other hand, for fixed contributions, the polynomial in question can be expressed

Xt^n + Y(t^n - 1)/(t-1)

which, when evaluated as t = 1+z gives

X(1+z)^n + Y((1+z)^n-1)/z

as the amount that one will have after n time intervals have elapsed.