Future Value of an Annuity (Regular Deposits)

Put the same amount away at the end of every period and each deposit compounds for a different length of time. The first sits longest, the last earns nothing at all, and adding up that staircase gives $\mathit{FV} = D[(1+i)^n - 1]/i$. Three annual deposits of $1,000 at 5% grow to $1000(1.1025) + 1000(1.05) + 1000 = 3152.50$, which the closed form reproduces exactly.

Read backwards, the same relation is the sinking fund: the deposit you need now to have a known sum later. A contractor who knows a $60,000 truck needs replacing in seven years can ask what monthly transfer gets there, rather than discovering the answer when the old one dies. This is how equipment reserves, roof funds and condominium capital plans are actually built.

The version here assumes deposits at the end of each period, the ordinary annuity. If you deposit at the beginning instead, every dollar earns one extra period, and the whole answer is simply larger by a factor of $(1+i)$.