# Sequences

> Mathematics · CIE IGCSE 0580 (2025-2027)
> Source: https://www.owlsprep.com/study/cie-0580-u2-sequences/

This guide covers all sequence content for CIE IGCSE Maths 0580 Core and Extended tiers, including term-to-term rules, continuing sequences, and nth term calculation for all tested sequence types.

**Prerequisites:** [Basic algebra substitution and rearrangement](https://www.owlsprep.com/study/cie-0580-u2-algebra-basics/); [Square, cube and integer powers](https://www.owlsprep.com/study/cie-0580-u1-number-powers-roots/)

## Learning objectives

- Continue linear, quadratic, cubic and exponential sequences using term-to-term rules
- Derive the nth term of linear, simple quadratic and simple cubic sequences (Core)
- Derive the nth term of exponential sequences and combined sequences, use Tₙ subscript notation (Extended)
- Verify nth term formulas against given sequence values

## Core: Term-to-Term Rules & Continuing Sequences

**Term-to-term rule** — A rule that tells you how to get from one term in a sequence to the next consecutive term

*Example:* For sequence 2, 5, 8, 11, the term-to-term rule is add 3

To continue a sequence, first identify the pattern between consecutive terms. For simple sequences this may be adding, subtracting, multiplying or dividing by a constant, or a more complex pattern for non-linear sequences.

**Worked example:** Continue the sequence 3, 7, 11, 15, __, __

1. First find the term-to-term rule: 7 - 3 = 4, 11 - 7 = 4, so the rule is add 4 each time
2. Calculate the next term: 15 + 4 = 19
3. Calculate the following term: 19 + 4 = 23
4. Final continued sequence: 3, 7, 11, 15, 19, 23

**Check your understanding**

1. What is the next term in the sequence 16, 8, 4, 2, __?

   *Why:* The term-to-term rule is divide by 2, so 2 ÷ 2 = 1

## Core: Nth Term of Linear Sequences

**Linear sequence** — A sequence where the difference between consecutive terms is constant (also called an arithmetic sequence)

*Notation:* Constant difference = d

To find the nth term of a linear sequence: 1. Calculate the constant difference d between terms. 2. Multiply d by n to get $dn$. 3. Find the constant c that you add to $dn$ to get the first term (when n=1). The nth term is written as $dn + c$.

**Worked example:** Find the nth term of the sequence 5, 9, 13, 17...

1. Calculate the constant difference d: 9 - 5 = 4, so d=4

   $$d=4$$
2. Start with $4n$. For n=1, $4(1) = 4$, but the first term is 5

   $$4(1)=4$$
3. Find c: 5 - 4 = 1, so c=1

   $$c=1$$
4. Final nth term: $4n + 1$. Test for n=2: $4(2) +1 =9$, which matches the sequence

   $$T_n = 4n + 1$$

> **Exam tip:** Always test your nth term formula against the first 3 terms of the sequence to confirm it works before using it for later terms.

## Core: Nth Term of Simple Quadratic Sequences

**Quadratic sequence** — A sequence where the second difference between consecutive terms is constant, and the nth term contains an $n^2$ term

*Notation:* Nth term form: $an^2 + bn + c$, $a \neq 0$

For simple quadratic sequences, the coefficient $a$ of $n^2$ is half of the constant second difference. For example, if the second difference is 2, $a=1$; if the second difference is 4, $a=2$.

**Worked example:** Find the nth term of the sequence 2, 5, 10, 17, 26...

1. Calculate first differences: 5-2=3, 10-5=5, 17-10=7, 26-17=9
2. Calculate second differences: 5-3=2, 7-5=2, 9-7=2, so the second difference is 2
3. Calculate a: $a = 2/2 = 1$, so $n^2$ is part of the nth term

   $$a=1$$
4. Subtract $n^2$ from each term: 2-1=1, 5-4=1, 10-9=1, the remainder is a constant 1
5. Final nth term: $n^2 + 1$. Test for n=3: $9 +1 =10$, which matches the sequence

   $$T_n = n^2 + 1$$

## Core: Nth Term of Simple Cubic Sequences

**Cubic sequence** — A sequence where the third difference between consecutive terms is constant, and the nth term contains an $n^3$ term

*Notation:* Nth term form: $an^3 + bn^2 + cn + d$, $a \neq 0$

For simple cubic sequences, the coefficient $a$ of $n^3$ is the third difference divided by 6. Most simple cubic sequences tested in 0580 are just $n^3$ with a constant added or subtracted.

**Worked example:** Find the nth term of the sequence 0, 7, 26, 63, 124...

1. Calculate first differences: 7-0=7, 26-7=19, 63-26=37, 124-63=61
2. Calculate second differences: 19-7=12, 37-19=18, 61-37=24
3. Calculate third difference: 18-12=6, 24-18=6, so the third difference is 6
4. Calculate a: $a = 6/6 = 1$, so $n^3$ is part of the nth term

   $$a=1$$
5. Subtract $n^3$ from each term: 0-1=-1, 7-8=-1, 26-27=-1, remainder is -1
6. Final nth term: $n^3 - 1$. Test for n=4: $64 -1 =63$, which matches the sequence

   $$T_n = n^3 -1$$

## Extended Adds: Exponential Sequences & Subscript Notation

**Exponential sequence** — A sequence where each term is multiplied by a constant common ratio $r$ to get the next term, and the nth term contains a power of $r$

*Notation:* Nth term form: $ar^{n}$ or $ar^{n-1}$, $a,r$ are constants

Extended tier requires you to use subscript notation $T_n$ to represent the nth term of a sequence, and find nth terms of sequences that combine multiple types, e.g. linear + quadratic.

**Worked example:** Find the nth term of the sequence 3, 6, 12, 24, 48... using $T_n$ notation.

1. Find the common ratio r: 6 ÷ 3 = 2, 12 ÷ 6 = 2, so r=2

   $$r=2$$
2. For n=1, $T_1 = 3 = 3 \times 2^{0}$, so the nth term is $3 \times 2^{n-1}$

   $$T_n = 3 \times 2^{n-1}$$
3. Test for n=3: $3 \times 2^{2} =12$, which matches the sequence

## Common pitfalls

- **Wrong:** Treating quadratic sequences as linear by using a constant first difference
  - Why it fails: Quadratic sequences have constant second differences, not first differences, so this leads to incorrect nth term formulas
  - Correct: If first differences are not constant, calculate second differences to check for a quadratic sequence
- **Wrong:** Using sum formulas for arithmetic/geometric series
  - Why it fails: Sum formulas are not part of the 0580 syllabus, so you will waste time and get no marks for using them
  - Correct: Only calculate individual terms or nth terms as required by the question
- **Wrong:** Identifying exponential sequences as linear by using addition instead of multiplication
  - Why it fails: Exponential sequences have a common ratio, not a common difference, and grow much faster than linear sequences
  - Correct: If terms grow very quickly, calculate the common ratio (divide consecutive terms) instead of a common difference
- **Wrong:** Testing nth term formulas only against the first term of the sequence
  - Why it fails: A formula that works for the first term may not work for later terms
  - Correct: Test your nth term against at least the first 3 terms of the sequence to confirm it is correct
- **Wrong:** Using Σ notation to write sums of sequences
  - Why it fails: Σ notation is not part of the 0580 syllabus, and examiners will not award marks for it
  - Correct: Write out terms individually if you need to add them, as required by the question

## Cheatsheet

| Sequence Type | How to Identify | Nth Term Rule | Tier |
| --- | --- | --- | --- |
| Linear | Constant first difference | $dn + c$, d = first difference | Core |
| Quadratic | Constant second difference | $an^2 + ...$, $a = ½$(second difference) | Core |
| Cubic | Constant third difference | $an^3 + ...$, $a = ⅙$(third difference) | Core |
| Exponential | Constant common ratio | $ar^n$, r = common ratio | Extended |
| Subscript notation | $T_n$ = nth term | Used for all sequence types | Extended |

## What's next

Now that you have mastered sequences for CIE IGCSE Maths 0580, you can move on to related algebra and graphs topics that often appear alongside sequence questions in exams. Practice applying your nth term calculation skills to past paper questions to build speed and confidence, and make sure you can distinguish between linear, quadratic, cubic and exponential sequences quickly under exam conditions. If you are studying Extended tier, make sure you are comfortable using $T_n$ notation in your answers as it is often required for higher-mark sequence questions.

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