Geometric & Numeric Patterns
Geometric and numeric patterns are sequences of numbers that follow specific rules for generating consecutive terms. In an arithmetic sequence, each term is found by adding a constant difference to the previous term, whilst in a geometric sequence, each term is found by multiplying the previous term by a constant ratio. These patterns appear throughout mathematics, from simple number sequences in Year 6 to complex series in A-level Further Maths.
Why it matters
Pattern recognition forms the foundation for understanding functions, exponential growth, and mathematical modelling. Arithmetic sequences model linear situations like monthly savings plans where £50 is deposited each month, creating the sequence 50, 100, 150, 200. Geometric sequences model exponential growth such as bacterial reproduction, where a population doubles every hour: 100, 200, 400, 800 bacteria. In finance, compound interest follows geometric patterns — £1000 invested at 5% annual interest becomes £1050, £1102.50, £1157.63 over successive years. These concepts underpin GCSE topics including quadratic sequences and A-level work on binomial expansions and calculus. Understanding patterns also develops logical reasoning skills essential for computer programming and data analysis.
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.
Worked examples
Is the sequence 5, 7, 9, 11, 13 arithmetic or geometric?
Answer: arithmetic
- Check differences between consecutive terms → 2, 2, 2, 2 — Differences: 2, 2, 2, 2. These are constant, so it is arithmetic.
- Check ratios between consecutive terms → 1, 1, 1, 1 — Ratios: 1, 1, 1, 1. These are not constant.
- State the answer → arithmetic (common difference d = 2) — The sequence is arithmetic with common difference d = 2.
In the sequence 3, 6, 12, 24, what is the common ratio?
Answer: 2
- Divide the second term by the first term → 6 ÷ 3 = 2 — 6 ÷ 3 = 2.
- Verify with another pair of terms → 12 ÷ 6 = 2 — 12 ÷ 6 = 2. The ratio is constant.
- State the common ratio → r = 2 — The common ratio is 2. Each term is multiplied by 2.
A geometric sequence starts 2, 4, 8, ... What is the 6th term?
Answer: 64
- Identify a₁ and r → a₁ = 2, r = 2 — The first term is 2. The common ratio is 4 ÷ 2 = 2.
- Write the nth term formula → aₙ = a₁ × rⁿ⁻¹ — The nth term of a geometric sequence is aₙ = a₁ × r^(n-1).
- Substitute n = 6 → a⁶ = 2 × 2⁵ = 2 × 32 = 64 — a_6 = 2 × 2⁵ = 2 × 32 = 64.
Common mistakes
- Confusing arithmetic and geometric patterns when ratios appear constant due to coincidence, such as incorrectly identifying 2, 4, 6 as geometric with ratio 2 instead of arithmetic with difference 2.
- Calculating the common ratio incorrectly by dividing the first term by the second, giving 3 ÷ 6 = 0.5 for the sequence 3, 6, 12 instead of the correct ratio 6 ÷ 3 = 2.
- Using the wrong position in the nth term formula, writing a₅ = 2 × 2⁵ = 64 instead of a₅ = 2 × 2⁴ = 32 for the sequence 2, 4, 8, 16.