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§ Expressions & Algebra

Algebraic Patterns

§ Expressions & Algebra

Algebraic Patterns

CCSS.4.OACCSS.5.OA3 min read

Algebraic patterns are sequences of numbers that follow a specific mathematical rule, such as adding a constant value or multiplying by a fixed ratio. The sequence 5, 8, 11, 14, 17 follows the rule 'add 3', whilst 2, 6, 18, 54 follows 'multiply by 3'. These patterns form the foundation for understanding algebraic expressions and formulae in KS3 and GCSE mathematics.

§ 01

Why it matters

Algebraic patterns appear throughout mathematics and real-world scenarios. Population growth models use geometric patterns where quantities multiply by consistent ratios — for example, bacteria doubling every 20 minutes follows the pattern 100, 200, 400, 800. Financial calculations rely on arithmetic patterns: monthly savings of £50 create the sequence £50, £100, £150, £200. In GCSE Foundation tier, students encounter linear sequences with nth term formulae like 3n + 2, whilst Higher tier explores quadratic patterns. Engineering uses these concepts for calculating stress loads, whilst computer science applies pattern recognition in algorithms. Understanding these sequences develops logical reasoning skills essential for advanced mathematics, from calculus to statistics, where patterns underpin mathematical modelling and prediction.

§ 02

How to solve algebraic patterns

Patterns & nth Term

  • Find the common difference (d) between consecutive terms.
  • nth term of a linear sequence: a + (n−1)d, or simplify to dn + c.
  • Check by substituting n = 1, 2, 3 to verify.
  • For non-linear: look at second differences.

Example: Sequence 3, 7, 11, 15: d=4 → nth term = 4n − 1.

§ 03

Worked examples

Beginner§ 01

What comes next? 10, 15, 20, 25, 30, __

Answer: 35

  1. Find the pattern → +5 — Each number increases by 5.
  2. Add 5 to the last term → 35 — 30 + 5 = 35.
Easy§ 02

What comes next? 15, 19, 23, 27, __

Answer: 31

  1. Find the common difference → +4 — 19 − 15 = 4. The rule is add 4.
  2. Add 4 to 27 → 31 — 27 + 4 = 31.
Medium§ 03

Find the rule and the next 2 terms: 3, 9, 15, 21, __, __

Answer: 27, 33

  1. Find the common difference → +6 — 9 − 3 = 6. The rule is +6.
  2. Find the 5th term → 27 — 21 + 6 = 27.
  3. Find the 6th term → 33 — 27 + 6 = 33.
§ 04

Common mistakes

  • Confusing additive and multiplicative patterns — writing that 2, 6, 18, 54 adds 4, then 12, then 36, instead of recognising the ×3 multiplicative rule
  • Incorrectly finding the common difference by subtracting in the wrong order — calculating 7 − 11 = −4 instead of 11 − 7 = 4 for the sequence 3, 7, 11, 15
  • Applying the wrong formula structure — writing 4n + 1 instead of 4n − 1 for the sequence 3, 7, 11, 15, leading to incorrect terms like 5, 9, 13, 17
§ 05

Frequently asked questions

What is the difference between arithmetic and geometric patterns?
Arithmetic patterns add or subtract the same value each time (5, 8, 11, 14 adds 3), whilst geometric patterns multiply or divide by the same value each time (3, 12, 48, 192 multiplies by 4). Arithmetic patterns have constant differences between consecutive terms, geometric patterns have constant ratios.
How do you find the nth term formula for a linear sequence?
Find the common difference (d) between consecutive terms, then use the formula a + (n−1)d, where a is the first term. For sequence 7, 11, 15, 19: d = 4, so nth term = 7 + (n−1)4 = 4n + 3. Check by substituting n = 1, 2, 3.
What does it mean when a pattern has second differences?
Second differences occur in quadratic sequences where the first differences aren't constant. For sequence 1, 4, 9, 16, 25, the first differences are 3, 5, 7, 9 (not constant), but second differences are 2, 2, 2 (constant). This indicates a quadratic pattern involving n².
How can you tell if a sequence is increasing or decreasing?
Compare consecutive terms: if each term is larger than the previous one, the sequence increases (2, 5, 8, 11). If each term is smaller, it decreases (20, 17, 14, 11). The common difference is positive for increasing sequences, negative for decreasing sequences.
Can patterns have negative numbers or decimals?
Yes, patterns can include negative numbers (−10, −7, −4, −1 adds 3) and decimals (1.5, 3.0, 4.5, 6.0 adds 1.5). The same rules apply: find the common difference or ratio between consecutive terms to identify the pattern and continue the sequence.
§ 06

See also

TermRatioMean
§ 06

Related topics

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