H2 Maths Sequences and Series | Free Notes

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 $u_n$ and $S_n$.
• 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.

Concrete example: In a savings recurrence, $u_{n+1}=1.005u_n+200$ 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 $u_{1}, u_{2}, \dots$ defined by either an explicit formula $u_n$ or a recurrence relation $u_{n+1} = f(u_n)$

Arithmetic progressions (AP)

• Common difference $d = u_{n+1} - u_n$.
• General term $u_n = u_{1} + (n - 1)d$

Geometric progressions (GP)

• Common ratio $r = \dfrac{u_{n+1}}{u_n}$ (assuming $u_n \neq 0$

Term-or-sum checkpoint

Before choosing a formula, decide what the question is actually asking for.

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 $\pu{200}$ monthly into an account earning $0.5\%$ interest per month. The balance after $n$ months obeys $u_{n+1} = 1.005 u_n + 200$

1. Generate the first few values on the GC using the recurrence mode to observe the growth.
2. Closed form after $n$ months:
   Result: $u_n = 200 \times \dfrac{1.005^n - 1}{0.005}$

Example 2 -- Sum to infinity

Given the GP $5, 2.5, 1.25, \dots$, determine $S_\infty$.

1. $u_{1} = 5$, $r = 0.5$.
2. Since $|0.5| \lt 1$, the sum converges.
3. $S_\infty = \frac{5}{1 - 0.5} = 10$

Example 3 -- Mixed AP/GP question

The first, fourth, and seventh terms of an AP coincide with the first three terms of a GP.

1. Let the common starting value be $a$.
2. AP terms: $a$, $a + 3d$, $a + 6d$. GP terms: $a$,

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 $S_n$ 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 $|r| \lt 1$ before using $S_\infty$.
• 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 $u_{n+1} > u_n$

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 $S_n$ and $S_\infty$ 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.

Next step: keep following the H2 Maths notes hub into Sub-topic 2.1 - Sequences & Series foundations for proofs, convergence diagnostics, and calculator workflows.

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 $S_\infty$ when $|r| \geq 1$: The infinite sum formula $S_\infty = a/(1-r)$

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 ($\sum k$, $\sum k^2$) 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.