H2 Maths Sequences & Series Formula Sheet | AP, GP & Sigma

Study guideUpdated 21 Aug 2026

H2 Maths sequences and series formula sheet: arithmetic and geometric progressions, sum to n terms, sum to infinity, sigma notation, and method of differences - every key result...

Q: What does H2 Maths Notes (JC 1-2): 2.1) Sequences and Series cover?
A: Convergence tests, AP/GP formulae, sigma notation, and standard H2 sequences problem types.
Download: Get the H2 Maths Sequences and Series formula sheet (PDF) for quick revision, or the complete notes (PDF) for the full walkthrough.
Before you revise
Keep a separate page for identities: arithmetic/geometric sums, binomial expansions, and standard limits. Most errors come from mixing up first term and common ratio definitions.
  • A sequence lists terms; a series adds them: Decide whether the question asks for unu_n or SnS_n.
  • AP uses difference; GP uses ratio: Identify aa, dd

Concrete example: For 5,2.5,1.25,…5, 2.5, 1.25, \dots, the ratio is 0.50.5. Since ∣r∣<1\lvert r\rvert \lt 1, the infinite sum is 51−0.5=10\frac{5}{1-0.5}=10

Status: SEAB's current H2 Mathematics (9758) syllabus PDF is labelled for 2026. Topic 2.1 scope is within Section A Pure Mathematics, which is assessed in Paper 1 (100 marks) and Paper 2 Section A (40 marks).

Formulas at a glance

Every result the 9758 syllabus expects you to recall, on one screen. The AP and GP sum formulas are not in MF27, so commit them to memory. The standard sigma identities (∑k\sum k, ∑k2\sum k^2) also need to be known. Worked examples for each appear in the sections below.

Arithmetic progression

QuantityFormula
General termun=a+(n−1)du_n = a + (n-1)d
Sum to nn

Geometric progression

QuantityFormula
General termun=arn−1u_n = ar^{n-1}
Sum to nn termsSn=a,1−rn1−rS_n = a,\dfrac{1 - r^n}{1 - r}

Sigma & standard results

IdentityFormula
Sum of integers∑k=1nk=n(n+1)2\displaystyle\sum_{k=1}^n k = \dfrac{n(n+1)}{2}

Method of differences (telescoping)

Term shapeDecomposition
1k(k+1)\dfrac{1}{k(k+1)}1k−1k+1\dfrac{1}{k} - \dfrac{1}{k+1}

Core Definitions

  • A sequence is an ordered list of terms un {u_n} . The nth partial sum is Sn=u1+u2+⋯+un S_n = u_1 + u_2 + \dots + u_n

Convergence and Limits

  • Use limits to test behaviour: lim⁡n→∞un \lim\limits_{n \to \infty} u_n exists if terms settle to a finite value.
  • For recursive sequences un+1=f(un) u_{n+1} = f(u_n)

Example -- Recurrence limit

Given u1=2 u_1 = 2 and un+1=12(un+5un) u_{n+1} = \frac{1}{2}\bigl(u_n + \frac{5}{u_n}\bigr)

  1. Suppose un→L u_n \to L ; then L=12(L+5L) L = \frac{1}{2}\bigl(L + \frac{5}{L}\bigr)

Recurrence proof checkpoint

For a recurrence question, separate the candidate limit from the proof that the sequence actually reaches it. Solving L=f(L)L = f(L) only gives a possible limit.

StepWhat to writeWhy it matters
Candidate limitLet un→Lu_n \to L, substitute into L=f(L)L = f(L), and choose the root consistent with the term sign.This identifies the value the sequence could approach.

Common trap: do not stop after solving L=f(L)L = f(L). A fixed point is not a convergence proof unless monotonicity and boundedness have also been established.

Sigma Notation and Manipulation

  • Express sums as ∑k=1nuk \sum\limits_{k=1}^n u_k . Familiar identities: ∑k=1nk=n(n+1)2,∑k=1nk2=n(n+1)(2n+1)6,∑k=1nrk=r1−rn1−r. \sum_{k=1}^n k = \frac{n(n + 1)}{2}, \qquad \sum_{k=1}^n k^2 = \frac{n(n + 1)(2n + 1)}{6}, \qquad \sum_{k=1}^n r^k = r \frac{1 - r^n}{1 - r}.

Sigma index-shift checkpoint

When changing the index, track the old index, the new index, the new limits, and the rewritten term together. Do not change only the lower limit and leave the term untouched.

Original sumNew index choiceNew limitsRewritten term
∑k=1nuk+1\sum\limits_{k=1}^n u_{k+1}Let j=k+1j = k + 1

Common trap: changing the index name does not change the sum, but changing a start or end value without changing the term usually changes the sum. Recalculate both limits from the same substitution before using a standard identity.

Telescoping decomposition checkpoint

For rational terms, aim to rewrite each term as "next difference" so the middle terms cancel when expanded.

Term shapeTry this rewriteCancellation checkCommon trap
1k(k+1)\dfrac{1}{k(k+1)}1k−1k+1\dfrac{1}{k}-\dfrac{1}{k+1}

Worked check: 1k(k+2)=12(1k−1k+2)\dfrac{1}{k(k+2)}=\dfrac{1}{2}\left(\dfrac{1}{k}-\dfrac{1}{k+2}\right)

Misconception check: telescoping is not just "cancel everything in the middle". First prove the decomposition, then expand enough terms to see exactly which first and last terms remain.

Example -- Telescoping sum

Evaluate the following series:

∑k=1n1k(k+1) \sum\limits_{k=1}^n \dfrac{1}{k(k + 1)}

  1. Decompose:
    Result: 1k(k+1)=1k−1k+1 \dfrac{1}{k(k + 1)} = \dfrac{1}{k} - \dfrac{1}{k + 1}

Arithmetic and Geometric Applications

  • Mixture problems: combine AP and GP components for salaries, payments, or depreciation.
  • Sum to infinity: verify ∣r∣<1 \lvert r \rvert \lt 1 before using S∞ S_\infty .
  • Insert means: For inserting k k arithmetic means between a a

AP or GP modelling checkpoint

Before choosing a formula, translate the wording into "add the same amount" or "multiply by the same factor".

Wording clueModel to useFirst labels to writeCommon trap
"increases by 150\pu{150} each year"APFirst term aa, common difference d=150d=\pu{150}, number of terms nn.

Worked check: a scholarship top-up starts at 500\pu{500} and increases by 40\pu{40} each semester for 6 semesters. This is an AP because the same amount is added each time, so a=500a=\pu{500}, d=40d=\pu{40}, and n=6n=6

Misconception check: percentage change is multiplicative, but fixed dollar change is additive. The units in the wording usually tell you whether the model is AP or GP.

Example -- Investment

A fund receives 200 \pu{200} monthly with constant growth of 1.2% 1.2\%. After 36 deposits, the total amount is an increasing annuity: S36=200×(1.012)36−10.012≈8.94×103. S_{36} = 200 \times \frac{(1.012)^{36} - 1}{0.012} \approx \pu{8.94 \times 10^3}.

Binomial Links

  • Use binomial coefficients: (nk)=n!k!(n−k)! \binom{n}{k} = \frac{n!}{k!(n - k)!}

Calculator Workflow

  • GC SEQ mode generates recursive terms quickly; verify monotonicity.
  • Use sum( or Σ( functions to compute partial sums while showing the manual formula in working.
  • Store ratio/difference in variables to avoid transcription errors.

Exam Watch Points

  • Always state whether a GP converges before quoting S∞ S_\infty .
  • Provide exact forms (fractions, radicals) rather than rounded decimals for general expressions.
  • In recurrence proofs, justify both monotonicity and boundedness; do not state convergence without proof.
  • Translate word problems into algebra carefully, labelling first term, common difference/ratio, and number of terms.

Practice Quiz

Revisit core AP/GP formulas, recurrence behaviour, and annuity applications with focused questions.

Quick Revision Checklist

  • Derive AP/GP formulae and apply them to contextual problems.
  • Manipulate sigma notation confidently, including telescoping and index shifts.
  • Analyse convergence of recursive sequences using fixed points.
  • Combine AP and GP reasoning in financial or growth questions.

Stay aligned with the full syllabus via the H2 Maths notes hub and continue with Topic 3 - Vectors once this sequence toolkit is secure.

Want weekly guided practice on Sequences and Series? Our H2 Maths tuition programme builds fluency in this topic through structured problem sets and exam-style drills.

Common exam mistakes

  • Applying S∞S_\infty without checking ∣r∣<1\lvert r\rvert < 1: The infinite geometric sum formula only converges when ∣r∣<1\lvert r\rvert < 1. Students frequently skip the convergence check and apply S∞=a/(1−r)S_\infty = a/(1-r)

Frequently asked questions

Are sequences and series in Paper 1 or Paper 2?
Topic 2 is Pure Mathematics and can appear in Paper 1 (100 marks) or Paper 2 Section A (40 marks). Financial modelling and recurrence questions are common long-structured questions.

Do I need to memorise the sum formulas for AP and GP?
Yes. The AP and GP sum formulas are not provided in the SEAB MF27 formula booklet, so they must be memorised. The standard sigma identities (∑k=n(n+1)/2\sum k = n(n+1)/2, ∑k2\sum k^2) also need to be known.

How do I prove a recursive sequence converges?
The standard approach for H2 Maths is: (1) assume the limit LL exists and solve L=f(L)L = f(L) for the fixed point; (2) prove the sequence is monotonic (via the sign of un+1−unu_{n+1} - u_n

Is there a formula sheet for H2 Maths sequences and series?
Yes - the "Formulas at a glance" section near the top of this page collects the AP and GP general terms, sums, standard sigma identities, and method-of-differences decompositions used in this topic. MF27 provides binomial and Maclaurin expansions, but it does not provide the AP or GP sum formulas, the standard sigma identities, or a table of standard sequence limits.

Other H2 Maths formula sheets

Revising more than one topic? Grab the matching one-page formula sheet:

For the official SEAB reference booklet, see the H2 Maths MF27 formula list.

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Marcus Pang
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Marcus Pang·Managing Director (Maths)