Series

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Use the expansion of (a + b)^n, where n is a positive integer, including the notations (n r) and n!
  2. Recognise arithmetic and geometric progressions
  3. Use the formulae for the nth term and for the sum of the first n terms to solve problems involving arithmetic or geometric progressions
  4. Use the condition for the convergence of a geometric progression, and the formula for the sum to infinity of a convergent geometric progression

1. Binomial Expansion

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What is Binomial Expansion?

The binomial expansion is a method for expanding expressions like (a + b)^n, where n is a positive whole number (1, 2, 3, 4, ...).

For example:

  • (a + b)² = a² + 2ab + b²
  • (a + b)³ = a³ + 3a²b + 3ab² + b³

Instead of multiplying out brackets repeatedly, we can use a formula.

The Binomial Theorem

For any positive integer n:

(a + b)^n = a^n + (n 1)a^(n-1)b + (n 2)a^(n-2)b² + (n 3)a^(n-3)b³ + ... + b^n

Understanding the Notation

Factorial Notation (n!)

The symbol n! (read as "n factorial") means multiply all whole numbers from n down to 1.

Examples:

  • 4! = 4 × 3 × 2 × 1 = 24
  • 5! = 5 × 4 × 3 × 2 × 1 = 120
  • 3! = 3 × 2 × 1 = 6
  • 1! = 1
  • 0! = 1 (by definition)

Binomial Coefficient (n r)

The symbol (n r) (read as "n choose r") represents the binomial coefficient. It tells us how many ways we can choose r objects from n objects.

The formula is:

(n r) = n! / [r! × (n - r)!]

Examples:

  • (4 2) = 4! / (2! × 2!) = 24 / (2 × 2) = 6
  • (5 3) = 5! / (3! × 2!) = 120 / (6 × 2) = 10

How to Expand (a + b)^n

The general term in the expansion is:

(n r) × a^(n-r) × b^r

where r goes from 0 to n.

Example: Expand (a + b)⁴

Using the formula:

  • When r = 0: (4 0)a⁴b⁰ = 1 × a⁴ × 1 = a⁴
  • When r = 1: (4 1)a³b¹ = 4 × a³ × b = 4a³b
  • When r = 2: (4 2)a²b² = 6 × a² × b² = 6a²b²
  • When r = 3: (4 3)a¹b³ = 4 × a × b³ = 4ab³
  • When r = 4: (4 4)a⁰b⁴ = 1 × 1 × b⁴ = b⁴

Therefore: (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴

Example: Expand (2x + 3)³

Here a = 2x, b = 3, n = 3

  • When r = 0: (3 0)(2x)³(3)⁰ = 1 × 8x³ × 1 = 8x³
  • When r = 1: (3 1)(2x)²(3)¹ = 3 × 4x² × 3 = 36x²
  • When r = 2: (3 2)(2x)¹(3)² = 3 × 2x × 9 = 54x
  • When r = 3: (3 3)(2x)⁰(3)³ = 1 × 1 × 27 = 27

Therefore: (2x + 3)³ = 8x³ + 36x² + 54x + 27

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