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§ Patterns

Geometric & Numeric Patterns

NO.LK20.93 min read

When Year 7 pupils encounter the sequence 3, 6, 12, 24, they often struggle to identify whether it's arithmetic or geometric. Recognising patterns in sequences forms a crucial foundation for GCSE algebra and helps students understand exponential growth in real-world contexts.

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§ 01

Why it matters

Geometric and arithmetic patterns appear everywhere in daily life, from compound interest calculations to population growth models. When students grasp that a savings account earning 5% annually follows geometric progression (£1000 becomes £1050, then £1102.50), they understand financial literacy concepts. Mobile phone data usage often doubles weekly during exam periods, creating geometric sequences. Even simple contexts like stacking paper (1 sheet = 0.1mm thick, 10 sheets = 1mm) help visualise arithmetic progressions. These patterns underpin GCSE topics including exponential functions, compound percentages, and series calculations. Students who master pattern recognition in Year 7 find algebraic manipulation significantly easier in Years 9-11, particularly when tackling quadratic sequences and geometric series.

§ 02

How to solve geometric & numeric patterns

Geometric & Numeric Patterns

  • Look at how each term relates to the previous: add, subtract, multiply, or divide?
  • For an arithmetic pattern, the difference between consecutive terms is constant.
  • For a geometric pattern, the ratio between consecutive terms is constant.
  • Write the rule, then use it to find the next terms.

Example: 2, 4, 8, 16, ... is geometric with ratio 2. Next term: 32.

§ 03

Worked examples

Beginner§ 01

Is the sequence 8, 10, 12, 14, 16 arithmetic or geometric?

Answer: arithmetic

  1. Check differences between consecutive terms → 2, 2, 2, 2 — Differences: 2, 2, 2, 2. These are constant, so it is arithmetic.
  2. Check ratios between consecutive terms → 1, 1, 1, 1 — Ratios: 1, 1, 1, 1. These are not constant.
  3. State the answer → arithmetic (common difference d = 2) — The sequence is arithmetic with common difference d = 2.
Easy§ 02

In the sequence 4, 12, 36, 108, what is the common ratio?

Answer: 3

  1. Divide the second term by the first term → 12 ÷ 4 = 3 — 12 ÷ 4 = 3.
  2. Verify with another pair of terms → 36 ÷ 12 = 3 — 36 ÷ 12 = 3. The ratio is constant.
  3. State the common ratio → r = 3 — The common ratio is 3. Each term is multiplied by 3.
Medium§ 03

A geometric sequence starts 5, 15, 45, ... What is the 6th term?

Answer: 1215

  1. Identify a₁ and r → a₁ = 5, r = 3 — The first term is 5. The common ratio is 15 ÷ 5 = 3.
  2. Write the nth term formula → aₙ = a₁ × rⁿ⁻¹ — The nth term of a geometric sequence is aₙ = a₁ × r^(n-1).
  3. Substitute n = 6 → a⁶ = 5 × 3⁵ = 5 × 243 = 1215 — a_6 = 5 × 3⁵ = 5 × 243 = 1215.
§ 04

Common mistakes

  • Students confuse arithmetic and geometric patterns, claiming 2, 6, 18, 54 has common difference 4 instead of common ratio 3.
  • When finding the 5th term of 3, 6, 12, pupils often write 3 × 2⁵ = 96 instead of using the correct formula 3 × 2⁴ = 48.
  • Students incorrectly calculate ratios by subtracting instead of dividing, claiming 4, 12, 36 has ratio 8 rather than 3.
  • Pupils forget to verify the pattern holds throughout, assuming 1, 3, 9, 28 is geometric when only the first three terms follow the rule.
Practice on your own
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§ 05

Frequently asked questions

How do I quickly distinguish arithmetic from geometric sequences?
Check differences first (arithmetic sequences have constant differences), then ratios (geometric sequences have constant ratios). For 2, 6, 18, 54: differences are 4, 12, 36 (not constant), but ratios are 3, 3, 3 (constant), so it's geometric.
What's the difference between sequences and series?
A sequence lists terms (2, 4, 8, 16, 32), whilst a series adds them (2 + 4 + 8 + 16 + 32 = 62). GCSE questions often ask for the sum of the first n terms, which requires series formulas rather than sequence formulas.
Why do geometric sequences grow so quickly?
Each term multiplies by the common ratio, creating exponential growth. Starting with 1 and doubling gives 1, 2, 4, 8, 16, 32, 64, 128. By the 10th term, you reach 512. This rapid growth explains compound interest and population explosions.
Can geometric sequences have negative ratios?
Yes, negative ratios create alternating sequences. With ratio -2, the sequence 3, -6, 12, -24, 48 switches signs each term. Students find these trickier but they appear regularly in GCSE papers, especially in coefficient problems.
How do I use the nth term formula correctly?
For geometric sequences, use aₙ = a₁ × r^(n-1), where a₁ is the first term and r is the common ratio. Remember the power is (n-1), not n. So the 5th term uses power 4, not 5.
§ 06

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