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 Video by Zeeshan Zamurred Sequences and Series

Edexcel A level Maths: 3.3 Geometric SequencesWatch the full walkthrough 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

Common ratio

Each term is the previous one multiplied by the $common ratio$ $r$ . You find $r$ by dividing any term by the one before it.

r = \frac{u _{n}}{u _{n - 1}}

Divide consecutive terms.

The nth term

The nth term is the first term times $r$ raised to $(n - 1)$ .

u_{n} = a r^{n - 1}

a

= first term,

r

= common ratio.

a = 2

r = 3

u_{6} = 2 \times 3^{5} = 2 \times 243

Answer

486

Finding n with logs

If you need the position $n$ 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

u_{n} = a r^{n - 1}

nth term.

r = \frac{u _{n}}{u _{n - 1}}

Common ratio.

r^{n - 1} = k \Rightarrow n 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.

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