Geometric Sequences and Series

By the end of this lesson, you will be able to find the $n$th term of a geometric sequence, compute partial sums of geometric series, and identify when a geometric series converges.

The Big Idea

A geometric sequence multiplies each term by a constant ratio $r$. Exponential functions restricted to integer inputs generate geometric sequences — making them the discrete version of exponential functions. They model population growth, compound interest, half-life, and viral spread.

Geometric Sequence

$n$th term: $a_n = a_1 \cdot r^{n-1}$

Common ratio: $r = \dfrac{a_2}{a_1} = \dfrac{a_3}{a_2}$ (constant)

Recursive: $a_n = r \cdot a_{n-1}$

Geometric Series (Partial Sum)

$$S_n = a_1 \cdot \frac{1 - r^n}{1 - r} \quad (r \neq 1)$$

If $r = 1$: $S_n = na_1$ (constant sequence — arithmetic with $d = 0$).

Derivation: $S_n = a_1 + a_1r + \cdots + a_1r^{n-1}$; multiply by $r$, subtract:

$(1-r)S_n = a_1(1-r^n)$

Example 1 — Finding the $n$th Term

Sequence: $3, 6, 12, 24, \ldots$. Find $a_8$.

$a_1 = 3$, $r = 2$

$a_8 = 3 \cdot 2^7 = 3 \cdot 128 = 384$

Example 2 — Computing a Sum

Sum the first 8 terms of $3, 6, 12, \ldots$

$S_8 = 3 \cdot \dfrac{1 - 2^8}{1 - 2} = 3 \cdot \dfrac{-255}{-1} = 765$

Example 3 — Decimal Ratio

Find $S_{10}$ for $100, 90, 81, 72.9, \ldots$

$r = 0.9$

$S_{10} = 100 \cdot \dfrac{1 - 0.9^{10}}{1 - 0.9} = 100 \cdot \dfrac{1 - 0.3487}{0.1} = 100 \cdot 6.513 \approx 651.3$

Example 4 — Finding $a_1$ and $r$

$a_3 = 12$ and $a_6 = 96$. Find $a_1$.

$\dfrac{a_6}{a_3} = r^3 = \dfrac{96}{12} = 8 \implies r = 2$

$a_3 = a_1 \cdot 4 = 12 \implies a_1 = 3$

Convergence Preview

A geometric series converges when $|r| < 1$ — the partial sums approach a finite limit as $n \to \infty$.

When $|r| < 1$: as $n \to \infty$, $r^n \to 0$, so:

$$S_\infty = \frac{a_1}{1 - r}$$

This is the infinite geometric series formula (next lesson).

Common Mistakes