Skip to content

Loading page…

© 2025 Eclat Institute. All rights reserved.

Empowering Singapore’s IP students to reach their fullest potential

@eclatinstitute on Instagram

Study Places Near Me

Open-source licences

Study SpotsStudy Places Near Me Blog Consult Us Now

Home
Blog
H2 Maths Notes
H2 Maths Notes 2 1 Sequences And Series

Loading article…

H2 Maths Notes (JC 1-2): 2.1) Sequences and Series

Download printable cheat-sheet (CC-BY 4.0)

07 Oct 2025, 00:00 Z

Join our Telegram study group

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.

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.

Core Definitions

A sequence is an ordered list of terms ${u_n}$ . The nth partial sum is $S_n = u_1 + u_2 + \dots + u_n$

Part 10 of 32

Previous topicH2 Maths Notes (JC 1-2): 1) Functions and Graphs Next topicH2 Maths Notes (JC 1-2): 2) Sequences and Series

Prefer a printable copy?

Download the full PDF cheatsheet

We keep the PDF version synced with this article so you can file it for revision alongside your notes.

.

Arithmetic progression (AP): constant difference

d

with

u_n = a + (n - 1)d

and

S_n = \tfrac{n}{2}\bigl(2a + (n - 1)d\bigr)

.

Geometric progression (GP): constant ratio

r

with

u_n = ar^{n - 1}

and

S_n = a \dfrac{1 - r^n}{1 - r}

for

r \neq 1

.

Infinite GP converges if

\lvert r \rvert < 1

with sum

S_\infty = \dfrac{a}{1 - r}

.

Convergence and Limits

Use limits to test behaviour: $\lim\limits_{n \to \infty} u_n$ exists if terms settle to a finite value.
For recursive sequences $u_{n+1} = f(u_n)$ , solve the fixed point $L = f(L)$ and show ${u_n}$ is monotonic and bounded.
Divergence occurs when terms grow without bound or oscillate without settling.

Example -- Recurrence limit

Given $u_1 = 2$ and $u_{n+1} = \frac{1}{2}\bigl(u_n + \frac{5}{u_n}\bigr)$ , show convergence.

Suppose $u_n \to L$ ; then $L = \frac{1}{2}\bigl(L + \frac{5}{L}\bigr)$ leading to $L^2 = 5$ so $L = \sqrt{5}$ (take positive root since terms stay positive).
Show $u_n$ is increasing and bounded above by $\sqrt{5}$ .

Sigma Notation and Manipulation

Express sums as $\sum\limits_{k=1}^n u_k$ . Familiar identities: $\sum_{k=1}^n k = \frac{n(n + 1)}{2}, \qquad \sum_{k=1}^n k^2 = \frac{n(n + 1)(2n + 1)}{6}, \qquad \sum_{k=1}^n r^k = r \frac{1 - r^n}{1 - r}.$
Split linear combinations: $\sum (a_k + b_k) = \sum a_k + \sum b_k$
To shift indices, substitute $k = j + c$ and adjust limits.

Example -- Telescoping sum

Evaluate the following series:

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

Decompose: $\dfrac{1}{k(k + 1)} = \dfrac{1}{k} - \dfrac{1}{k + 1}$
Terms telescope to $1 - \dfrac{1}{n + 1} = \dfrac{n}{n + 1}$

Arithmetic and Geometric Applications

Mixture problems: combine AP and GP components for salaries, payments, or depreciation.
Sum to infinity: verify $\lvert r \rvert < 1$ before using $S_\infty$ .
Insert means: For inserting $k$ arithmetic means between $a$ and $b$ , use AP formula; for geometric means, use GP ratio $r = \left(\dfrac{b}{a}\right)^{1/(k+1)}$ .

Example -- Investment

A fund receives $\pu{200}$ monthly with constant growth of $1.2\%$ . After 36 deposits, the total amount is an increasing annuity: $S_{36} = 200 \times \frac{(1.012)^{36} - 1}{0.012} \approx \pu{8.56 \times 10^3}.$

Binomial Links

Use binomial coefficients: $\binom{n}{k} = \frac{n!}{k!(n - k)!}$
Expand $(1 + x)^n$ and match coefficients to sequences.
Recognise common combinatorial sums like $\sum \dbinom{n}{k} = 2^n$

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 $S_\infty$ .
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.

H2 Maths Notes (JC 1-2): 2.1) Sequences and Series

.

Related Posts

H2 Biology Practical: Enzyme Kinetics (Catalase) Guide
H2 Biology Practical: Optical Microscope Mastery
H2 Biology Practical: Osmosis and Diffusion (Potato Cores & Agar Blocks)