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 interest, and exponential graphs.

What Is a Geometric Sequence?

A geometric sequence has a first term a and a common ratio r. Each term is obtained by multiplying the previous term by r.

For example, 3, 6, 12, 24, 48, … has a = 3 and r = 2 (each term is doubled).

Another example: 100, 50, 25, 12.5, … has a = 100 and r = 0.5 (each term is halved).

Key Formulas

Here, a is the first term, r is the common ratio, and n is the position of the term.

Step-by-Step Method

1. Identify the first term a from the sequence.
2. Find the common ratio r by dividing the second term by the first (or any consecutive pair).
3. Verify r is constant by checking another pair of consecutive terms.
4. Use the nth term formula a × r^(n−1) to find any term.
5. To find which term has a given value, set a × r^(n−1) equal to that value and solve for n.

Worked Example 1 — Foundation Level

Question: Find the common ratio and the next two terms of the sequence 5, 15, 45, 135, …

Working:

r = 15 ÷ 5 = 3. Check: 45 ÷ 15 = 3 ✓ and 135 ÷ 45 = 3 ✓

Next term = 135 × 3 = 405. Following term = 405 × 3 = 1215.

Answer: r = 3; next two terms are 405 and 1215

Worked Example 2 — Higher Level

Question: The first term of a geometric sequence is 4 and the common ratio is 0.5. Find the 7th term.

Working:

Using the nth term formula: nth term = a × r^(n−1).

7th term = 4 × 0.5^(7−1) = 4 × 0.5^6 = 4 × 0.015625 = 0.0625.

Answer: 0.0625

Worked Example 3 — Exam Style

Question: The 2nd term of a geometric sequence is 12 and the 5th term is 324. Find the common ratio and the first term.

Working:

2nd term: ar = 12. 5th term: ar⁴ = 324.

Divide the 5th term by the 2nd term: ar⁴ ÷ ar = r³ = 324 ÷ 12 = 27.

So r³ = 27, meaning r = 3.

From ar = 12: a × 3 = 12, so a = 4.

Check 5th term: 4 × 3⁴ = 4 × 81 = 324 ✓

Answer: r = 3, a = 4

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).

Practice Questions

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.

Practise geometric sequences with instant feedback free on GCSEMathsAI.

Related Topics

• Sequences and Nth Term
• Quadratic Sequences and Nth Term
• Growth and Decay

Summary

• A geometric sequence multiplies each term by a constant ratio r.
• The nth term formula is a × r^(n−1), where a is the first term.
• Find r by dividing any term by the previous term.
• When given two terms, divide them to find a power of r, then solve for r.
• Geometric sequences are closely related to exponential growth and decay problems.