Pearson Edexcel · A-Level · Mathematics
Course version: 9MA0 · Version from Mar 2026
Understand and work with geometric sequences and series, including the…
Full specification wording
Sequences and series: 4.5 Understand and work with geometric sequences and series, including the formulae for the nth term and the sum of a finite geometric series; the sum to infinity of a convergent geometric series, including the use of |r| < 1; modulus notation.
A geometric sequence multiplies by a constant common ratio r each time, so a, ar, ar², ar³, ...
Checked against the Pearson Edexcel 9MA0 specification (opens in a new tab) on .
- Point 5 of 6
- 4 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
A geometric sequence multiplies by a constant common ratio r each time, so a, ar, ar², ar³, ... The nth term is uₙ = ar^(n − 1). The sum of the first n terms is Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1, or equivalently a(rⁿ − 1)/(r − 1). When |r| < 1 the terms shrink towards zero and the series converges to S∞ = a/(1 − r); the modulus notation |r| < 1 means r lies strictly between −1 and 1. For example, 8, 4, 2, 1, ... has a = 8 and r = ½, so u₆ = 8 × (½)⁵ = ¼ and S∞ = 8/(1 − ½) = 16. Identify a and r, check convergence before using S∞, and solve equations involving powers, often using logarithms.
What earns the marks
- States uₙ = ar^(n − 1) and substitutes the correct a, r and n to find a term or form an equation.
- Uses Sₙ = a(1 − rⁿ)/(1 − r) for a finite geometric series, including when r is negative or fractional.
- Applies the convergence condition |r| < 1 and calculates S∞ = a/(1 − r) only when that condition holds.
- Finds a or r from given terms, for example by dividing two terms to eliminate a, and handles negative or fractional ratios.
- Solves equations involving rⁿ, using logarithms where necessary, and interprets whether a series converges or diverges.
Common mistakes and how to put them right
Using uₙ = arⁿ instead of ar^(n − 1).
Put it right by using the exponent n − 1.
Applying S∞ = a/(1 − r) when |r| ≥ 1.
Put it right by checking the modulus condition first and stating that the series diverges.
Dividing terms in the wrong order when finding r.
Put it right by dividing a later term by the previous term, uₙ₊₁ ÷ uₙ.
Exam tips
- State the value of r and check |r| < 1 before quoting a sum to infinity.
- When r is negative, keep the sign throughout and check whether terms alternate.
- For equations in n, take logarithms of both sides and round up if the context requires a whole number of terms.
Try the lesson
The start of the Sequences and series: 4.5 Understand and work with geometric sequences and series, including the formulae for the nth term and the sum of a finite geometric series; the sum to infinity of a convergent geometric series, including the use of |r| < 1; modulus notation. lesson, made from checked teaching notes for this specification point.
Have a go first
A geometric sequence has first term 3 and common ratio 2. What is the 6th term of this sequence?
Listen
A geometric sequence multiplies by a constant common ratio r each time, so a, ar, ar², ar³, ... The nth term is uₙ = ar^(n − 1). The sum of the first n terms is Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1, or equivalently a(rⁿ − 1)/(r − 1). When |r| < 1 the terms shrink towards zero and the series converges to S∞ = a/(1 − r); the modulus notation |r| < 1 means r lies strictly between −1 and 1. For example, 8, 4, 2, 1, ... has a = 8 and r = ½, so u₆ = 8 × (½)⁵ = ¼ and S∞ = 8/(1 − ½) = 16. Identify a and r, check convergence before using S∞, and solve equations involving powers, often using logarithms. A strong answer earns marks for these points. States uₙ = ar^(n − 1) and substitutes the correct a, r and n to find a term or form an equation. Uses Sₙ = a(1 − rⁿ)/(1 − r) for a finite geometric series, including when r is negative or fractional. Applies the convergence condition |r| < 1 and calculates S∞ = a/(1 − r) only when that condition holds. Finds a or r from given terms, for example by dividing two terms to eliminate a, and handles negative or fractional ratios. A common slip: Using uₙ = arⁿ instead of ar^(n − 1). Instead, using the exponent n − 1. Exam tip: State the value of r and check |r| < 1 before quoting a sum to infinity.
See it
Key words
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nth term formula
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Sum of a finite geometric series
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Convergence and sum to infinity
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Modulus notation
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See every definition
- nth term formula
- The nth term is uₙ = ar^(n − 1), found by substituting the correct a, r and n.
- Sum of a finite geometric series
- The sum of the first n terms is Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1, or equivalently a(rⁿ − 1)/(r − 1).
- Convergence and sum to infinity
- When |r| < 1 the terms shrink towards zero and the series converges to S∞ = a/(1 − r).
- Modulus notation
- The notation |r| < 1 means r lies strictly between −1 and 1.
Label the diagram
Some boxes have lost their names. Pick the right name for each number.
1Box 1
2Box 2
3Box 3
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