A geometric series adds the terms of a geometric sequence. Its sum formula uses a different trick from Arithmetic Series: multiply by $r$ and subtract. It is the math behind savings plans and loan payments.
Key idea
For $r \ne 1$: $$S_n = a + ar + ar^2 + \cdots + ar^{n-1} = \frac{a(1 - r^n)}{1 - r}.$$ Here $a$ is the first term and $n$ is the number of terms. If $r = 1$, every term is $a$ and $S_n = na$.
Why it works
Write $S = a + ar + \cdots + ar^{n-1}$ and multiply by $r$: $rS = ar + ar^2 + \cdots + ar^n$. Subtract: almost everything cancels, leaving $S - rS = a - ar^n$. Factor: $S(1 - r) = a(1 - r^n)$, and divide by $1 - r$ (which is why $r = 1$ needs its own formula). For a savings plan, each deposit earns interest for a different number of months, so the balance is a geometric series with ratio $1 + i$.
Worked examples
Worked example: A doubling sum
Find $3 + 6 + 12 + \cdots + 384$.
- $a = 3$, $r = 2$. How many terms? $384 = 3 \cdot 2^7$, so the exponents run from 0 to 7: $n = 8$.
- $S_8 = \frac{3(1 - 2^8)}{1 - 2} = \frac{3(-255)}{-1} = 765$.
Answer: $765$
Worked example: A fraction ratio
Find $64 + 32 + 16 + 8 + 4$.
- $a = 64$, $r = \frac{1}{2}$, $n = 5$.
- The formula: $$S_5 = \frac{64\left(1 - \frac{1}{32}\right)}{\frac{1}{2}} = 128 \cdot \frac{31}{32} = 124.$$
Answer: $124$
Worked example: Saving every month
You deposit \$100 at the end of each month at 6% per year compounded monthly ($i = 0.005$). Find the balance after 2 years.
- The balance is $100 + 100(1.005) + \cdots + 100(1.005)^{23}$, a geometric series with $a = 100$, $r = 1.005$, $n = 24$.
- $S = \frac{100(1.005^{24} - 1)}{0.005}$ $\approx 2543.20$.
Answer: about \$2,543.20 (you deposited \$2,400)
Common mistake
Using $n - 1$ instead of $n$, or applying the formula when $r = 1$. In $3 + 6 + \cdots + 384$ the last exponent is 7, but there are 8 terms. Count terms by listing exponents from 0. And if $r = 1$, the formula divides by 0: just multiply $a$ by $n$.