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3.3PureCore

Geometric Sequences

A geometric sequence multiplies by the same factor — the common ratio — each step. Its nth-term formula uses powers, and it models growth and decay just like exponentials.

25 min Sequences and Series
Edexcel A level Maths: 3.3 Geometric SequencesCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Identify a geometric sequence and its common ratio
  • Use the nth term formula arⁿ⁻¹
  • Find a missing term or ratio
  • Use logs to find the position n
1

Common ratio

Each term is the previous one multiplied by the common ratio\textbf{common ratio} rr. You find rr by dividing any term by the one before it.

r=unun−1r = \dfrac{u_{n}}{u_{n-1}}
Divide consecutive terms.
2

The nth term

The nth term is the first term times rr raised to (n−1)(n - 1).

un=arn−1u_n = ar^{n-1}
aa = first term, rr = common ratio.
Example 1Find the 6th term of 2, 6, 18, …
1a=2a = 2, r=3r = 3.
2u6=2×35=2×243u_6 = 2 \times 3^{5} = 2 \times 243.
Answer
486486
3

Finding n with logs

If you need the position nn of a term, the unknown ends up in a power, so take logs to solve — exactly the technique from Year 1 exponentials.

Tip — Term position n sits in an exponent ⟶ take logs of both sides to solve for n.

Formula recap

un=arn−1u_n = ar^{n-1}
nth term.
r=unun−1r = \dfrac{u_n}{u_{n-1}}
Common ratio.
rn−1=k⇒n via logsr^{n-1} = k \Rightarrow n \text{ via logs}
Finding the position.

Common mistakes to avoid

Adding the common difference instead of multiplying by the ratio.
Geometric sequences MULTIPLY by r each step.
Using rⁿ instead of rⁿ⁻¹.
The first term has r⁰; the nth term uses r^(n−1).

Key takeaways

  • Geometric: constant common ratio r (multiply each step).
  • nth term: uₙ = arⁿ⁻¹.
  • Find r by dividing consecutive terms; use logs for the position n.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. The nth term of a geometric sequence is:

    • arn−1ar^{n-1}
    • a+(n−1)da + (n-1)d
    • arnarn
    • a+nra + nr
    Show answer and explanation

    Answer: arn−1ar^{n-1}

    First term × r^(n−1).

  2. The common ratio of 2, 6, 18, … is:

    • 33
    • 22
    • 44
    • 66
    Show answer and explanation

    Answer: 33

    6 ÷ 2 = 3.

  3. The 6th term of 2, 6, 18, … is:

    • 486486
    • 162162
    • 243243
    • 729729
    Show answer and explanation

    Answer: 486486

    2 × 3⁵ = 486.

  4. You find the common ratio by:

    • Dividing consecutive terms
    • Subtracting terms
    • Adding terms
    • Squaring a term
    Show answer and explanation

    Answer: Dividing consecutive terms

    r = uₙ/uₙ₋₁.

  5. To find the position n of a term, you use:

    • Logarithms
    • The quadratic formula
    • Differentiation
    • Pythagoras
    Show answer and explanation

    Answer: Logarithms

    n is in the exponent.

  6. A sequence with a constant ratio is:

    • Geometric
    • Arithmetic
    • Linear
    • Periodic
    Show answer and explanation

    Answer: Geometric

    Constant common ratio.

Next step

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