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Sequences and series: A Level Maths knowledge organiser

Everything to know about sequences and series on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Arithmetic series
The sum of the terms of an arithmetic sequence, which goes up by d each time.
Geometric series
The sum of the terms of a geometric sequence, which is multiplied by r each time.
Sum to infinity
For a geometric series with |r| < 1, the sum gets closer to a/(1 − r).
Binomial expansion
Expanding (a + b)ⁿ using the coefficients ⁿCᵣ.

Key formulas

Formulas marked with a label are given in the exam (we only say so where our formula sheet confirms it). Learn the rest.

Arithmetic \(n\)th term\(u_n=a+(n-1)d\)
Arithmetic seriesIn the formula booklet: Edexcel, AQA, OCR\(S_n=\tfrac12n(a+l)=\tfrac12n\big[2a+(n-1)d\big]\)
Geometric \(n\)th term\(u_n=ar^{n-1}\)
Geometric seriesIn the formula booklet: Edexcel, AQA, OCR\(S_n=\frac{a(1-r^n)}{1-r},\) \(S_\infty=\frac a{1-r}\ (|r|<1)\)
Binomial, \(n\in\mathbb N\)In the formula booklet: Edexcel, AQA, OCR\((a+b)^n=a^n+\tbinom n1a^{n-1}b\) \({}+\tbinom n2a^{n-2}b^2+\cdots+b^n\)
Binomial coefficientIn the formula booklet: Edexcel, AQA, OCR\(\binom nr={}^nC_r=\frac{n!}{r!(n-r)!}\)
Binomial, \(|x|<1\), \(n\in\mathbb Q\)In the formula booklet: Edexcel, AQA, OCR\((1+x)^n=1+nx+\frac{n(n-1)}{1\times2}x^2+\cdots\)
\((a+bx)^n=a^n(1+\tfrac bax)^n\), valid for \(|x|<\left|\tfrac ab\right|\)
Sigma; recurrence\(\sum_{r=1}^nr=\tfrac12n(n+1);\) \(u_{n+1}=f(u_n)\)

Worked example

An arithmetic series has first term 7 and common difference 4. Find the sum of the first 20 terms.

  1. Sₙ = ½n[2a + (n − 1)d]
  2. S₂₀ = 10 × (14 + 19 × 4) = 10 × 90

Answer: S₂₀ = 900

Common mistakes

  • Binomial expansion (rational n): factor not raised to the power, validity not stated
  • Proving by checking a few examples
  • Exhaustion missing a case
  • 'let n = 3' as a general proof

More on what examiners see students get wrong: Examiner Insights.

You should be able to…

  • Use Pascal's triangle and factorial notation, including ⁿCᵣ = n!/(r!(n − r)!), to find binomial coefficients and count arrangements in simple cases.
  • Use the first few terms of a binomial expansion to estimate values such as 1.02⁸, and explain why the approximation works when x is small.
  • Use the nth term and sum formulae for geometric sequences and series, and solve problems for the number of terms with logarithms where needed.
  • Generate sequences from recurrence relations such as uₙ₊₁ = 2uₙ − 1, and find unknown constants from given terms of the sequence.
  • Model real situations such as savings, loans and population growth with arithmetic and geometric series, and comment on how realistic the models are.
  • Split an expression into partial fractions and then use binomial expansions of each part to expand it, stating the range of validity of the result.

The printable sheet

Sequences and series knowledge organiser for A Level Maths: one A4 page of key definitions, formulas, a worked example and common mistakes
Sequences and series knowledge organiser (A Level Maths), A4. Download the PDF.

Revise it next

Other A Level Maths topics: Proof · Algebra and functions · Coordinate geometry · Trigonometry · Exponentials and logarithms · Differentiation · Integration · Numerical methods · Vectors · Sampling and data · Probability · Binomial and normal distributions · Hypothesis testing · Kinematics · Forces and Newton's laws · Moments · All A Level Maths organisers