Sequences

Q: Do I need to learn sum formulas for sequences for 0580?

No, sum formulas for arithmetic or geometric series are not included in the CIE IGCSE 0580 syllabus and will not be tested. You only need to calculate individual terms and nth terms.

Mathematics· 2.7· 25 min read

1. Core: Term-to-Term Rules & Continuing Sequences★★☆☆☆⏱ 5 min

📘 Definition

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

✓ Quick check

What is the next term in the sequence 16, 8, 4, 2, __?
Reveal answer
1 —
The term-to-term rule is divide by 2, so 2 ÷ 2 = 1

2. Core: Nth Term of Linear Sequences★★☆☆☆⏱ 7 min

📘 Definition

Linear sequence

Constant difference = d

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

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

📐 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 $4 n$ . 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: $4 n + 1$ . 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.

3. Core: Nth Term of Simple Quadratic Sequences★★★☆☆⏱ 7 min

📘 Definition

Quadratic sequence

Nth term form: $a n^{2} + bn + c$ , $a \neq = 0$

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

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
$T_n = n^2 + 1$

4. Core: Nth Term of Simple Cubic Sequences★★★★☆⏱ 6 min

📘 Definition

Cubic sequence

Nth term form: $a n^{3} + b n^{2} + c n + d$ , $a \neq = 0$

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

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
$T_n = n^3 -1$

5. Extended Adds: Exponential Sequences & Subscript Notation★★★★☆Extended only⏱ 5 min

📘 Definition

Exponential sequence

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

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 $r$

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

6. Common Pitfalls

Wrong move:

Treating quadratic sequences as linear by using a constant first difference

Why:

Quadratic sequences have constant second differences, not first differences, so this leads to incorrect nth term formulas

Correct move:

If first differences are not constant, calculate second differences to check for a quadratic sequence

Wrong move:

Using sum formulas for arithmetic/geometric series

Why:

Sum formulas are not part of the 0580 syllabus, so you will waste time and get no marks for using them

Correct move:

Only calculate individual terms or nth terms as required by the question

Wrong move:

Identifying exponential sequences as linear by using addition instead of multiplication

Why:

Exponential sequences have a common ratio, not a common difference, and grow much faster than linear sequences

Correct move:

If terms grow very quickly, calculate the common ratio (divide consecutive terms) instead of a common difference

Wrong move:

Testing nth term formulas only against the first term of the sequence

Why:

A formula that works for the first term may not work for later terms

Correct move:

Test your nth term against at least the first 3 terms of the sequence to confirm it is correct

Wrong move:

Using Σ notation to write sums of sequences

Why:

Σ notation is not part of the 0580 syllabus, and examiners will not award marks for it

Correct move:

Write out terms individually if you need to add them, as required by the question

7. Quick Reference Cheatsheet

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

8. Frequently Asked

Do I need to learn sum formulas for sequences for 0580?

No, sum formulas for arithmetic or geometric series are not included in the CIE IGCSE 0580 syllabus and will not be tested. You only need to calculate individual terms and nth terms.

Can I use Tₙ notation in Core tier answers?

You can use Tₙ notation in any answer, but it is only a required skill for Extended tier questions.

Going deeper

official resourceCIE IGCSE 0580 Syllabus (2025-2027)Official syllabus for sequence content

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.