Nobody saves in one lump. They save $200 a month, and every instalment starts earning the day it lands — so the first deposit compounds for the whole term and the last one compounds for no time at all. Adding that staircase term by term works and takes all afternoon. The closed form is $\mathit{FV} = D\,\dfrac{(1+i)^n - 1}{i}$, read aloud as F V equals D times, one plus i to the n, minus one, over i.

Every letter, in words: $D$ is the deposit made at the end of each period, in dollars, and it is the same every time — that is what makes it an annuity. $i$ is the rate per period as a decimal. $n$ is the number of deposits, not the number of years, and the two only agree when deposits are annual. $\mathit{FV}$ is the balance immediately after the last deposit. Solve for $D$ instead and you get the sinking-fund question every municipality answers before it replaces a pump.

The bracket is doing real work. $(1+i)^n$ is what one dollar becomes; subtracting 1 keeps only the GROWTH; dividing by $i$ counts how many deposits' worth of growth that represents. Drop the − 1 and you have invented a principal nobody deposited. And the sanity rail before any arithmetic: the answer must sit above $D \times n$ — the deposits alone — and below what the whole pile would earn if it had all arrived on day one.