Sheet № 140 · Higher only · AQA · Edexcel · OCR
Geometric Sequences –
A geometric sequence is a sequence where each term is found by multiplying the previous term by a fixed number called the common ratio. Unlike arithmetic sequences (which add a constant), geometric sequences grow or shrink by a constant multiplier. This topic appears on the Higher paper and is closely linked to growth and decay, compound
§Key definitions
Question:
Find the common ratio and the next two terms of the sequence 5, 15, 45, 135, …
Answer:
r = 3; next two terms are 405 and 1215
Q1 (Foundation):
Write the first 5 terms of the geometric sequence with a = 2 and r = 4.
Q2 (Higher):
Find the 6th term of the geometric sequence 1000, 200, 40, 8, …
Q3 (Higher):
The 3rd term of a geometric sequence is 18 and the 6th term is 486. Find r and the first term.
§Formulas to memorise
nth term = a × r^(n−1)
Common ratio r = any term ÷ previous term
Identify the first term a — from the sequence.
Find the common ratio r — by dividing the second term by the first (or any consecutive pair).
Verify r is constant — by checking another pair of consecutive terms.
Use the nth term formula — a × r^(n−1) to find any term.
To find which term has a given value — , set a × r^(n−1) equal to that value and solve for n.
Worked example
Find the common ratio and the next two terms of the sequence 5, 15, 45, 135, …
Working:
⚠ Common mistakes
- ✗Confusing arithmetic and geometric sequences. Arithmetic sequences add a constant; geometric sequences multiply by a constant. Check whether the difference or the ratio between consecutive terms is constant.
- ✗Using the wrong exponent in the formula. The nth term uses r^(n−1), not r^n. The first term (n = 1) must give a × r⁰ = a.
- ✗Forgetting negative or fractional ratios. A sequence like 2, −6, 18, −54 has r = −3. A sequence like 80, 20, 5, 1.25 has r = 0.25.
✦ Exam tips
- →If a sequence alternates between positive and negative, the common ratio is negative.
- →When r is between −1 and 1 (exclusive), the terms get smaller and approach zero — this links to convergent series.
- →Always verify your common ratio with at least two pairs of consecutive terms.
- →Geometric sequences connect to compound interest (r = 1 + rate) and depreciation (r = 1 − rate).