Mathematics 9709 · syllabus 1.6
Cambridge A Level Mathematics 9709 — Series topical past papers
The binomial expansion of (a + b)ⁿ for a positive whole number n, and arithmetic and geometric progressions, including sums to infinity.
- Questions
- 222
- Years
- 2009–2026
- With mark scheme
- 98%
- Questions 2022–2026
- 72
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What Series covers
Series in Paper 1 (section 1.6) has two strands. The first is the binomial expansion of (a + b)ⁿ when n is a positive whole number, using the coefficients ⁿCr. Questions ask for the first few terms, a particular coefficient, or the term independent of x in an expansion with x and 1/x, and often multiply an expansion by another bracket and ask for one coefficient of the product.
The second strand is progressions. In an arithmetic progression each term is the previous one plus a common difference d; in a geometric progression each term is the previous one times a common ratio r. You use the formulae for the nth term and for the sum of the first n terms, and the sum to infinity of a geometric progression, which exists only when |r| < 1. Questions often link two progressions that share terms.
The binomial expansion for any rational power, (1 + x)ⁿ with |x| < 1, is Paper 3 algebra rather than part of this section. Progressions are often set in context, such as savings, distances run or lengths cut from a rope, where you must first decide which kind of progression the situation describes.
In the syllabus
The headings this topic has in the Cambridge International 2026–2027 syllabus, as the board prints them.
- 1.6 Series
What you should be able to do
- Expand (a + b)ⁿ for a positive integer n using ⁿCr
- Find a particular term or coefficient, including the term independent of x
- Find a coefficient in the product of a binomial expansion and another bracket
- Recognise arithmetic and geometric progressions and find a, d or r
- Use the nth-term and sum formulae, and the sum to infinity when |r| < 1
- Set up and solve simultaneous equations from information about two progressions
Key ideas
- (a + b)ⁿ = aⁿ + ⁿC₁aⁿ⁻¹b + ⁿC₂aⁿ⁻²b² + … + bⁿ, where ⁿCr = n! / (r!(n − r)!)
- Arithmetic: uₙ = a + (n − 1)d and Sₙ = ½n(2a + (n − 1)d) = ½n(a + l), where l is the last term
- Geometric: uₙ = arⁿ⁻¹ and Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1
- Sum to infinity S∞ = a/(1 − r), valid only when |r| < 1
- a, b, c are in arithmetic progression if 2b = a + c, and in geometric progression if b² = ac
How Series has been examined
From 2022 to 2026, Series was on Paper 1 · Pure Mathematics 1 in 11 of 11 exam series, worth about 18% of that paper’s marks.
| Paper | Questions | Share of marks | Series |
|---|---|---|---|
| Paper 1 · Pure Mathematics 1 | 72 | 18% | 11 of 11 |
Momentum’s own count, over the questions in its bank for these years: a series is one exam session, every zone included, and a question filed under two chapters counts in both. Median marks per question: 6.
Filed in Momentum under: Binomial Theorem, Sequences & Series.
A worked example
Find the term independent of x in the expansion of (2x − 1/x²)⁶.
- The general term is ⁶Cr × (2x)^(6 − r) × (−1/x²)^r.
- Its power of x is (6 − r) − 2r = 6 − 3r.
- The term is independent of x when 6 − 3r = 0, so r = 2.
- That term is ⁶C₂ × 2⁴ × (−1)² = 15 × 16 × 1.
Answer: 240
Written by Momentum for this page, with its own numbers; it is not taken from a past paper.
Mistakes that cost marks
- Not raising the whole second term to its power: in (2 − 3x)⁵ the x² term is ⁵C₂ × 2³ × (−3x)², and writing −3x² instead of (−3x)² = 9x² gives the wrong coefficient.
- Losing the sign of a negative term: odd powers of (−3x) are negative.
- Using uₙ = a + nd, which is the (n + 1)th term: the nth term is a + (n − 1)d.
- Quoting a sum to infinity without checking |r| < 1: if the ratio is outside that range, the series has no sum to infinity.
- Confusing the nth term with the sum of n terms, for example giving the distance run on day 20 when the question asks for the total over 20 days.
Exam technique
- Write the general term ⁿCr aⁿ⁻ʳbʳ and simplify its power of x before choosing the term you want; this is the quickest route to a term independent of x.
- In progression questions, write down the two equations the information gives in a and d (or a and r) before solving; dividing one geometric equation by another cancels a.
- Give a coefficient as a number: 'the coefficient of x²' wants the number in front of x², not the whole term.
Practising in Momentum
- Every question in a chapter as one list, from every paper and year in the bank, newest first or shuffled.
- Each question turns over to its own part of the mark scheme, where the board published one.
- Narrow by paper, year, session, zone and marks, and star the questions worth keeping.
- Pick chapters, a number of questions and a time, and the exam builder draws a mock to sit and mark.
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