Geometric sequences GCSE questions ask you to find, extend, or describe a sequence where each term is multiplied by a fixed common ratio to get the next. Once you can identify the ratio and apply the nth term formula arⁿ⁻¹, you can jump straight to any term without listing the whole sequence.
What is a geometric sequence?
A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a fixed value called the common ratio, usually written r. Unlike an arithmetic sequence, where you add a constant difference, a geometric sequence grows — or shrinks — by a constant factor.
For example, 3, 6, 12, 24, 48 is geometric: each term is double the one before, so r = 2. The sequence 100, 50, 25, 12.5 is also geometric, with r = 0.5 — the terms shrink rather than grow, but the multiplying rule still applies.
How do you find the common ratio?
Divide any term by the term immediately before it. Using consecutive terms is essential — dividing non-adjacent terms will not give the correct value.
r = term₂ ÷ term₁
Worked example: find the common ratio of 5, 15, 45, 135.
- Divide the second term by the first: 15 ÷ 5 = 3.
- Check with another pair: 45 ÷ 15 = 3.
- Check again: 135 ÷ 45 = 3.
Since every pair gives the same value, r = 3. If the ratio calculated from different pairs of terms does not match, the sequence is not geometric.
What is the formula for the nth term of a geometric sequence?
The nth term of a geometric sequence is given by:
Uₙ = a × rⁿ⁻¹
where a is the first term, r is the common ratio, and n is the term number.
Steps to build the formula from a given sequence:
- Identify the first term, a.
- Find the common ratio, r, by dividing consecutive terms.
- Substitute a and r into Uₙ = a × rⁿ⁻¹.
Worked example: find the nth term of 4, 8, 16, 32, 64.
- First term: a = 4.
- Common ratio: r = 8 ÷ 4 = 2.
- nth term: Uₙ = 4 × 2ⁿ⁻¹.
Verification: n = 1: 4 × 2⁰ = 4 × 1 = 4 ✓. n = 3: 4 × 2² = 4 × 4 = 16 ✓. n = 5: 4 × 2⁴ = 4 × 16 = 64 ✓.
How do you find a specific term using the formula?
Once you have Uₙ = a × rⁿ⁻¹, substitute the term number directly — there is no need to write out the whole sequence.
Worked example: using Uₙ = 4 × 2ⁿ⁻¹ (from above), find the 8th term.
- Substitute n = 8: U₈ = 4 × 2⁷.
- Calculate 2⁷ = 128.
- U₈ = 4 × 128 = 512.
This is far faster than doubling repeatedly from the fourth term onward, and it becomes essential once term numbers reach double figures.
How are geometric sequences different from arithmetic sequences?
The two sequence types are tested side by side at GCSE, and mixing them up is one of the most common errors.
| Feature | Arithmetic sequence | Geometric sequence |
|---|---|---|
| Rule between terms | Add a constant difference, d | Multiply by a constant ratio, r |
| nth term formula | Uₙ = a + (n − 1)d | Uₙ = a × rⁿ⁻¹ |
| Example | 2, 5, 8, 11, 14 (d = 3) | 2, 6, 18, 54, 162 (r = 3) |
| Growth pattern | Straight-line (linear) growth | Curved (exponential) growth |
A fast way to tell them apart: check whether consecutive terms have a constant difference (arithmetic) or a constant ratio (geometric). If neither works cleanly, check second differences — the sequence may be quadratic instead.
How do you find a missing term in a geometric sequence?
When one term is missing from the middle of a sequence, use the terms either side of the gap.
- Divide the term after the gap by the term before the gap to find r² (since two multiplying steps have been skipped).
- Take the square root to find r, remembering that r could be negative.
- Multiply the term before the gap by r to find the missing term.
Worked example: find the missing term in 3, ?, 27.
- 27 ÷ 3 = 9, so r² = 9.
- r = 3 (taking the positive root).
- Missing term = 3 × 3 = 9.
Check: 3, 9, 27 — each term is three times the one before. ✓
Frequently asked questions
Can the common ratio of a geometric sequence be negative?
Yes. A negative common ratio makes the sequence alternate between positive and negative terms. For example, 2, −6, 18, −54 has r = −3: multiplying by a negative number flips the sign each time while the magnitude still grows by a factor of 3. The nth term formula Uₙ = a × rⁿ⁻¹ still applies exactly as normal.
Can the common ratio be a fraction?
Yes, and this produces a decreasing sequence when the fraction is between 0 and 1. For example, 80, 40, 20, 10 has r = 0.5, or equivalently ½. Each term is found by multiplying, not dividing, so writing r as a fraction rather than switching to division keeps the formula consistent.
How do you know if a sequence is geometric and not arithmetic?
Check whether dividing consecutive terms gives a constant value. If term₂ ÷ term₁ equals term₃ ÷ term₂ equals term₄ ÷ term₃, the sequence is geometric. If instead subtracting consecutive terms gives a constant value, the sequence is arithmetic. A sequence cannot usually satisfy both tests unless the common difference is zero.
What happens to a geometric sequence when r is between 0 and 1?
The terms get progressively smaller and approach — but never quite reach — zero. For example, with a = 16 and r = 0.5, the sequence runs 16, 8, 4, 2, 1, 0.5, continuing to halve indefinitely. This decreasing pattern is still fully geometric, and the same Uₙ = a × rⁿ⁻¹ formula finds any term in it.
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