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06062025–2027 syllabus

ADDITIONAL MATHEMATICS · CHAPTER 12

Series

Use a general term to target binomial powers and use progression structure to model finite or infinite sums.

Additional Mathematics4 connected sectionsSyllabus-aligned guide

LEARNING OBJECTIVES

What you will be able to do

  • expand positive-integer binomials
  • use the general binomial term
  • recognise and solve arithmetic progression problems
  • use finite and infinite geometric series

AT A GLANCE

Syllabus0606Coverage2025–2027Sections4LevelAdditional Mathematics

INTRODUCTION · THE BIG IDEA

Use a general term to target binomial powers and use progression structure to model finite or infinite sums.

A sequence lists terms; a series adds them. Additional Mathematics focuses on binomial expansions and on arithmetic or geometric progressions with predictable term and sum formulas.

Targeted questions rarely require a whole expansion. A general term identifies the exact power or constant term efficiently.

01

SECTION 01

Binomial expansion

Core concept

The term indexed by r is term r+1 because r begins at zero. Preserve signs inside b, especially when expanding (a − b)ⁿ.

RULE 1
Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳbʳ, 0 ≤ r ≤ n
Original worked example

Find one coefficient without expanding

  1. Find the coefficient of x² in (3x − 2)⁵.
  2. Use Tᵣ₊₁ = ⁵Cᵣ(3x)⁵⁻ʳ(−2)ʳ.
  3. Require 5 − r = 2, so r = 3.
  4. Coefficient = ⁵C₃ × 3² × (−2)³.

Answer: −720.

02

SECTION 02

Arithmetic progressions

Core concept

An arithmetic progression has constant difference d. Move from the first term a to the nth term by adding d exactly n−1 times.

RULE 1
uₙ = a + (n−1)d
RULE 2
Sₙ = n/2[2a + (n−1)d]
RULE 3
Sₙ = n(a+l)/2
Arithmetic symbols
SymbolMeaning
afirst term
dcommon difference
uₙnth term
llast term
Sₙsum of first n terms
03

SECTION 03

Geometric progressions

Core concept

A geometric progression has constant ratio r. Each term is formed by multiplying the previous one by r.

DETAILED EXPLANATION

  • A ratio between 0 and 1 gives decreasing positive magnitudes.
  • Use exact fractional ratios where possible.
  • Context may restrict n to positive integers.
RULE 1
uₙ = arⁿ⁻¹
RULE 2
Sₙ = a(1−rⁿ)/(1−r), r ≠ 1
ORIGINAL STUDY DIAGRAMIdentify a progression
1Compare consecutive differences
2Compare consecutive ratios
3Choose AP or GP formulas
4Solve for parameters
5Check the requested term or sum
04

SECTION 04

Convergence and sum to infinity

Core concept

A geometric progression converges only when repeated multiplication makes rⁿ approach zero. This happens precisely when |r| < 1.

RULE 1
S∞ = a/(1−r), valid only when |r| < 1
Original worked example

Evaluate a recurring geometric total

  1. A series begins 12 + 6 + 3 + …
  2. Identify a = 12 and r = 1/2.
  3. Since |r| < 1, a sum to infinity exists.
  4. Use 12/(1 − 1/2).

Answer: S∞ = 24.

QUICK CHAPTER SUMMARY

The ideas to carry forward

  • The binomial general term targets a required power.
  • An AP has constant difference.
  • A GP has constant ratio.
  • Finite sum formulas depend on the progression type.
  • A geometric sum to infinity exists only for |r| < 1.

QUICK REVISION CHECKLIST

Can you do each of these without your notes?

  • expand positive-integer binomials
  • use the general binomial term
  • recognise and solve arithmetic progression problems
  • use finite and infinite geometric series
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