Q: What does H2 Maths Notes (JC 1-2): 2) Sequences and Series cover? A: Arithmetic and geometric progressions, convergence tests, sigma notation, and calculator workflows for modelling recurrence relations.
Before you start Refresh IP-level arithmetic and geometric sequences. Revisit the MOE definitions so your notation matches the mark scheme and set your graphing calculator (GC) to Exam Mode before drilling.
Terms and sums are different objects: Label un and Sn.
Recurrence questions need the next-term rule: Generate a few terms before generalising.
Modelling questions need interpretation: State what the limit, sum, or divergence means in context.
In a savings recurrence, un+1=1.005un+200 means the old balance grows first, then the new deposit is added. That order matters when forming the closed form.
Status: SEAB's current H2 Mathematics (9758) syllabus PDF is labelled for 2026. Topic 2 expectations are within Section A Pure Mathematics, which is assessed in Paper 1 (100 marks) and Paper 2 Section A (40 marks).
2.1 | Language and Notation
Core ideas
A sequence is an ordered list u1,u2,… defined by either an explicit formula un or a recurrence relation un+1=f(un).
A series is the sum of the first n terms: Sn=u1+u2+⋯+un
Use sigma notation Sn=∑k=1nuk to condense working and match marking scheme expectations.
Arithmetic progressions (AP)
Common difference d=un+1−un.
General term un=u1+(n−1)d.
Sum Sn=2n(u1+un)
Geometric progressions (GP)
Common ratio r=unun+1 (assuming un=0).
General term un=u1rn−1.
Finite sum Sn=u11−r1−rn
Infinite sum converges only if ∣r∣<1, giving S∞=1−ru1
Term-or-sum checkpoint
Before choosing a formula, decide what the question is actually asking for.
Wording clue
Quantity wanted
First setup
Common trap
"Find the 20th term"
One term, u20
Use the AP or GP term formula.
Using Sn because the word "series" appears nearby.
"Total after 20 deposits"
Partial sum, S20
Decide whether the first deposit grows for 19 or 20 periods.
Off-by-one errors in finance recurrences.
"Long-run total" or "sum to infinity"
Infinite sum, S∞
Check ∣r∣<1 before substituting.
Applying S∞
"Given un+1 in terms of un"
Recurrence output or limiting behaviour
Generate terms, then solve the recurrence if needed.
Treating a recurrence like an explicit formula immediately.
Worked check: if a savings question asks for the "total amount in the account after 24 monthly deposits", label whether the first deposit earns interest for 23, 24, or 0 months before writing the sum. The formula changes when the deposit is made at the start or end of each month.
2.2 | Worked Examples
Example 1 -- Savings plan (recurrence)
A student deposits 200 monthly into an account earning 0.5% interest per month. The balance after n months obeys un+1=1.005un+200 with u0=0.
Generate the first few values on the GC using the recurrence mode to observe the growth.
Closed form after n months: Result:un=200×0.0051.005n−1 (finite geometric sum of monthly deposits).
There is no finite limiting balance because the multiplier exceeds 1; the account grows without bound as n→∞.
Example 2 -- Sum to infinity
Given the GP 5,2.5,1.25,…, determine S∞.
u1=5, r=0.5.
Since ∣0.5∣<1, the sum converges.
S∞=1−0.55=10.
Example 3 -- Mixed AP/GP question
The first, fourth, and seventh terms of an AP coincide with the first three terms of a GP.
Let the common starting value be a.
AP terms: a, a+3d, a+6d. GP terms: a, ar, ar2.
Equate: a+3d=ar and a+6d=ar2. Eliminating d
For non-zero a, the only consistent ratio is r=1 so d=0; both sequences are constant (a,a,a,…)
2.3 | Calculator Workflows
Use GC recurrence mode (SEQ on TI, RECUR on Casio) to generate terms quickly and observe convergence.
Employ the summation feature to evaluate Sn without manual addition, but record the command in your working (e.g. sumSeq(200(1.005)^(X-1), X, 1, 24)).
For sigma notation, store n as a variable and test values so you can spot algebraic errors before finalising.
2.4 | Exam Watch Points
Always state convergence conditions ∣r∣<1 before using S∞.
In recurrence modelling, quote the balance to the nearest cent and interpret the result (e.g. long-run savings, population cap).
Justify inequalities when proving monotonic sequences un+1>un.
Combine AP and GP formulas carefully-write down both before substituting values.
Practice Quiz
Consolidate AP/GP identities, convergence tests, and sigma manipulations before moving to sub-topic drills.
2.5 | Quick Revision Checklist
Translate context into explicit or recurrence form fluently.
Compute Sn and S∞ with and without the GC.
Explain what convergence means in words (e.g. limiting value for savings).
Manipulate sigma notation expressions by splitting or re-indexing the sum.
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
Using S∞ when ∣r∣≥1: The infinite sum formula S∞=a/(1−r) only converges when ∣r∣<1. Applying it without checking this condition - or when r=1 exactly - results in a nonsensical answer.
Mixing up un and Sn: The nth term formula gives the value of a single term, not the accumulated sum. A question asking for "the total after n
Off-by-one errors in the number of terms: For the series from um to un, the number of terms is n−m+1
Forgetting to justify convergence in recurrence proofs: Stating that a sequence converges without showing it is both monotonic and bounded is incomplete. Both conditions must be verified explicitly.
Sign errors in AP/GP when d or r is negative: A negative common difference or ratio changes the direction of inequalities in monotonicity proofs. Write out the first few terms to confirm the sign pattern before generalising.
Frequently asked questions
Is Topic 2 (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). Expect financial modelling questions that combine AP/GP with real-world contexts.
Can I use the GC to find terms and partial sums? Yes. The GC SEQ or RECUR mode is useful for checking your analytic answers. However, you must still show the formula and working - simply quoting a calculator output without method steps will not earn full marks.
Do I need to memorise the standard sum formulas, or are they given? The AP and GP sum formulas are not provided in the SEAB formula list, so they must be memorised. Standard sigma identities (∑k, ∑k2) are also not on the reference sheet.
Sources
SEAB H2 Mathematics syllabus (9758), examinations from 2026 - Topic 2 Sequences and series, including arithmetic/geometric sequences, sigma notation, convergence, and recurrence relations, in Section A Pure Mathematics. See the citation entry above for the official PDF URL.
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. Always count carefully before substituting into the sum formula.
