Number Patterns and Sequences

Number patterns follow rules. This lesson covers arithmetic sequences (add/subtract a constant), geometric sequences (multiply by a constant), and how to find missing terms and formulas.

What Is a Sequence?

A sequence is an ordered list of numbers following a rule. Each number is called a term.

$2, 5, 8, 11, 14, \ldots$

Arithmetic Sequences

In an arithmetic sequence, the same value is added (or subtracted) each time. This value is the common difference $d$ .

$d = \text{term}_{n+1} - \text{term}_n$

Example: 3, 7, 11, 15, 19 → $d = 4$

nth Term — Arithmetic

a_n = a_1 + (n-1) \cdot d

a₁ = first term, d = common difference, n = term number.

Geometric Sequences

In a geometric sequence, each term is multiplied by the same value — the common ratio $r$ .

$r = \frac{\text{term}_{n+1}}{\text{term}_n}$

Example: 2, 6, 18, 54 → $r = 3$

Other Patterns

Square numbers: 1, 4, 9, 16, 25 … ( $n^2$ )
Triangle numbers: 1, 3, 6, 10, 15 … ( $\frac{n(n+1)}{2}$ )
Fibonacci: 1, 1, 2, 3, 5, 8, 13 … (each term = sum of two previous)

Tip To spot the pattern, look at differences between consecutive terms first. If differences are constant → arithmetic. If ratios are constant → geometric.

1
Find consecutive differences
$5-2=3, 8-5=3, 11-8=3$ → arithmetic, $d=3$
2
Use the formula
$a_n = 2 + (n-1) \times 3$
3
Find the 10th term
$a_{10} = 2 + 9 \times 3 = 29$

Find the 8th term of the sequence 5, 10, 20, 40 ...

$r = 2$ (geometric), $a_1 = 5$

$a_8 = 5 \times 2^7 = 5 \times 128 = 640$

Real world: phone tariff plans

A mobile tariff charges €10 per month plus €0.05 per minute of calls. The total cost follows the arithmetic sequence starting at €10 with each minute adding €0.05. After 60 minutes: $10 + 60 \times 0.05 = 13\text{ EUR}$ . Recognising the arithmetic rule helps consumers compare tariff deals.

Real world: saving money over time

A student saves €20 in January, then increases savings by €5 each month: 20, 25, 30, 35 … This is an arithmetic sequence ( $d = 5$ ). The 12th term (December) $= 20 + 11 \times 5 = 75\text{ EUR}$ , so they know exactly how much they will be saving by year's end.

Real world: bacteria growth

A bacterium divides every hour, doubling in number: 1, 2, 4, 8, 16 … This geometric sequence ( $r = 2$ ) is used in medicine to model infection spread and calculate how long before a colony reaches dangerous levels.

Tip Arithmetic sequences grow steadily; geometric sequences grow explosively. Recognising which type you have determines which formula to use.

Practice Problems

What is the 15th term of the arithmetic sequence 4, 7, 10, 13, ...?

Show answer

a₁=4, d=3. a₁₅ = 4 + 14×3 = 4 + 42 = 46.

Find the missing term: 3, __, 27, 81.

Show answer

r = 3. Missing term = 9.

Is 200 a term in the sequence 5, 10, 15, 20, ...?

Show answer

aₙ = 5n. 5n = 200 → n = 40. Yes, the 40th term.