# Geometric Sequences and Series

> Patterns that multiply by the same number each step. The nth term, sums, and the limiting sum of an infinite series.

From HSC Maths Advanced on WhizzNotes, free study notes for NSW students: https://whizznotes.com/mathematics-advanced/geometric-sequences/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The bouncing ball
Kai drops a tennis ball from 2 m. Each bounce reaches 60% of the height before. 2 m, then 1.2 m, then 0.72 m, then 0.432 m... The bounces get smaller and smaller but, in theory, never quite stop.

Here is the surprise. Even with endless bounces, the total distance the ball travels is a **fixed, finite number**. Patterns where you multiply by the same number each step are **geometric sequences**. They describe bouncing balls, compound interest and the spread of a virus.

**In short:** A geometric sequence multiplies by the same ratio $r$ each time, like 2, 6, 18, 54. Its $n$th term is $T_n = ar^{n-1}$, and when $-1 < r < 1$ the infinite sum is $S_\infty = \dfrac{a}{1 - r}$.

## Starting point: Years 9 and 10

### Multiplying patterns

In 3, 6, 12, 24, ..., each term is **double** the one before. To find the multiplier, divide a term by the one before it: $\dfrac{6}{3} = 2$.

In 80, 40, 20, 10, ..., the multiplier is $\dfrac{40}{80} = \dfrac{1}{2}$, so the terms halve.

### Calculator tip: List terms with Ans
Type the first term and press `=`, then type `× 3` (the common ratio) and keep pressing `=`. Each press multiplies the last answer by 3. For a percentage increase of 5%, multiply by 1.05, not 0.05.

### Adding or multiplying?

| Sequence | Same difference? | Same ratio? | Type |
|---|---|---|---|
| 5, 9, 13, 17 | yes, $+4$ | no | arithmetic |
| 5, 10, 20, 40 | no | yes, $\times 2$ | geometric |
| 1, 4, 9, 16 | no | no | neither (square numbers) |

### Repeated percentage change

Growing by 10% means multiplying by 1.1 each step. A 1000 dollar investment growing 10% a year gives 1000, 1100, 1210, 1331, ..., a geometric sequence with multiplier 1.1.

### Worked example: Doubling bacteria
A colony of 50 bacteria doubles every hour. After 1 hour: 100. After 2: 200. After $n$ hours: $50 \times 2^n$. After 6 hours: $50 \times 64 = 3200$.

### Watch out: Ratio is next divided by previous
The ratio is $\dfrac{T_2}{T_1}$, not $\dfrac{T_1}{T_2}$. For 12, 4, $\dfrac{4}{3}$, ... the ratio is $\dfrac{1}{3}$.

**Mind map**

```text
Geometric sequences
  Common ratio r
    r = T₂ / T₁
    Multiply each step
  General term
    Tₙ = arⁿ⁻¹
    Exponential pattern
  Sum
    Sₙ = a(rⁿ - 1)/(r - 1)
  Infinite sum
    Needs -1 < r < 1
    S = a / (1 - r)
  Uses
    Interest, decay, bouncing
```

### Year 11: The general term
A **geometric sequence** has first term $a$ and **common ratio** $r$:

$$a, \; ar, \; ar^2, \; ar^3, \; \dots \qquad T_n = ar^{n-1}$$

**Example.** For 3, 6, 12, ..., $T_{10} = 3 \times 2^9 = 1536$.

**Which term?** Which term of 2, 6, 18, ... is 4374? Solve $2 \times 3^{n-1} = 4374$, so $3^{n-1} = 2187 = 3^7$ and $n = 8$. For awkward numbers, take logs.

**Three terms in a row** $x$, $y$, $z$ are geometric if $\dfrac{y}{x} = \dfrac{z}{y}$, so $y^2 = xz$.

**Finding $a$ and $r$ from two terms.** If $T_2 = 12$ and $T_4 = 48$, divide: $\dfrac{ar^3}{ar} = r^2 = 4$, so $r = 2$ or $r = -2$, and $a = 6$ or $a = -6$. The ratio can be negative: the signs then alternate.

**Solving for $n$ with logs.** A salary of 60 000 dollars rising 5% a year is $60\,000 \times 1.05^{n-1}$ in year $n$. To pass 80 000: $1.05^{n-1} > \frac{4}{3}$, so $n - 1 > \dfrac{\ln(4/3)}{\ln 1.05} \approx 5.9$, giving $n = 7$, the 7th year.

### Year 12: Geometric series and the limiting sum
The sum of the first $n$ terms is

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

or $\dfrac{a(1 - r^n)}{1 - r}$, which is easier when $r < 1$.

**Example.** $3 + 6 + 12 + \dots$ for 8 terms: $S_8 = \dfrac{3(2^8 - 1)}{1} = 765$.

**Limiting sum.** If $-1 < r < 1$, the terms shrink towards 0 and $r^n \to 0$. The sum of infinitely many terms approaches

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

**Kai's ball.** Down 2 m first. Then each bounce goes up and down: $2(1.2 + 0.72 + \dots)$. The bounces form a series with $a = 1.2$ and $r = 0.6$:

$$\text{total} = 2 + 2 \times \frac{1.2}{1 - 0.6} = 2 + 6 = 8 \text{ m}$$

**Recurring decimals.** $0.\overline{3} = 0.3 + 0.03 + 0.003 + \dots = \dfrac{0.3}{1 - 0.1} = \dfrac{1}{3}$.

If $|r| \geq 1$, the sum grows without limit and there is no limiting sum.

**When does a limiting sum exist?** If the ratio contains $x$, solve $-1 < r < 1$. For $1 + 2x + 4x^2 + \dots$, $r = 2x$, so $-\frac{1}{2} < x < \frac{1}{2}$. For $5 - 5x + 5x^2 - \dots = 12$: $\dfrac{5}{1 + x} = 12$, so $x = -\frac{7}{12}$, which is allowed because $|x| < 1$.

**Exponential terms.** $e^{-x} + e^{-2x} + \dots$ has $a = r = e^{-x}$, which is less than 1 for $x > 0$, so the sum is $\dfrac{e^{-x}}{1 - e^{-x}} = \dfrac{1}{e^x - 1}$.

**Long-run problems.** A dose $D$ is taken every day and 80% of the drug remains a day later. Just after a dose, the amount is $D + 0.8D + 0.8^2D + \dots$, which approaches $\dfrac{D}{1 - 0.8} = 5D$.

**Taking turns.** Ann and Ben take turns, Ann first; each try succeeds with probability $\frac{1}{3}$ for Ann and $\frac{1}{2}$ for Ben. Ann wins on her first try, or after both miss once, and so on: $\frac{1}{3} + \left(\frac{2}{3} \cdot \frac{1}{2}\right)\frac{1}{3} + \dots = \dfrac{1/3}{1 - 1/3} = \dfrac{1}{2}$.

## Quick summary

- A geometric sequence multiplies by the same ratio $r$ each step: $r = \dfrac{T_2}{T_1}$.
- $T_n = ar^{n-1}$.
- $S_n = \dfrac{a(r^n - 1)}{r - 1}$.
- If $-1 < r < 1$, the infinite sum is $S_\infty = \dfrac{a}{1 - r}$.
- Repeated percentage change gives a geometric sequence.

## Quiz

Try these questions from the chapter quiz.

1. What is the next term of 2, 6, 18, 54, ...?
   - A. 108
   - B. 162
   - C. 72
   - D. 216
   - Answer: B. 162. Multiply by 3.

2. What is the common ratio of 100, 50, 25, ...?
   - A. 2
   - B. $\dfrac{1}{2}$
   - C. $-50$
   - D. 25
   - Answer: B. $\dfrac{1}{2}$. $\dfrac{50}{100} = \dfrac{1}{2}$.

3. Which sequence is geometric?
   - A. 1, 3, 5, 7
   - B. 1, 3, 9, 27
   - C. 1, 4, 9, 16
   - D. 2, 4, 7, 11
   - Answer: B. 1, 3, 9, 27. Each term is 3 times the last.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/mathematics-advanced/geometric-sequences/#quiz