Add the terms of a geometric sequence and the total is $S_n = \dfrac{a\,(r^{n} - 1)}{r - 1}$ — read aloud: S-n equals a, times r-to-the-n minus one, over r minus one. For 3 + 6 + 12 + … (8 terms): $3 \times (256 - 1)/1 = 765$. Notice the shape: the LAST term, 384, beats everything before it combined, 381. Multiplying growth is back-loaded — that is its whole personality, and the reason late slices of it dominate every total.

Now hold a thought, because it is the hinge of this chapter: a savings plan — the same deposit every year, each one earning interest for a different number of years — is exactly a geometric series. Each deposit is a term; $1 + i$ is the ratio. Lesson eight cashes this sentence in at the bank.