This module is about turning a list of numbers into a rule. You will find the nth term of a linear sequence, test a rule before you trust it, tell arithmetic change from geometric change, use second differences for quadratic patterns, and separate a term’s position from its value.
Check the current Cambridge IGCSE Mathematics 0580 syllabus to see which sequence types are in your exam year and whether they sit in Core or Extended content. The method stays the same across versions. Our Mathematics learning guide shows where this module sits among the others.
What should you already know?
You need to substitute a number into an expression such as 4n + 1, and to solve a simple one-step or two-step equation. Negative numbers should feel comfortable, because decreasing sequences use them. If signs still slow you down, revisit number sense and exact arithmetic first.
An orienting example
The sequence 5, 9, 13, 17, … continues in the same way. Find the nth term, then find which term equals 101.
Step 1, differences: 9 − 5 = 4, 13 − 9 = 4, 17 − 13 = 4. The difference is constant, so the rule starts with 4n.
Step 2, adjust: 4n gives 4, 8, 12, 16. Each term needs 1 more, so the rule is 4n + 1.
Step 3, check: n = 1 gives 5 and n = 4 gives 17. Both match.
Step 4, position: set 4n + 1 = 101. Then 4n = 100, so n = 25. The value 101 is the 25th term.
Distant check: counting on from 5 in steps of 4 for 24 steps gives 5 + 96 = 101. Both routes agree.
That one question used most of the module: differences, a rule, a test, and the difference between “n” and “the value”.
In which order should you study it?
- Find a linear term rule from differences: the core method everything else builds on.
- Test a term rule against distant terms: stops a rule that fits three terms from failing on the tenth.
- Distinguish arithmetic from geometric change: tells you whether to add or to multiply.
- Use second differences to investigate a quadratic rule: extends the idea to patterns that curve.
- Explain the difference between term position and term value: fixes the most common wording error in these questions.
Then work through the mixed practice set. One lesson a day and the practice set at the weekend is a steady pace.
Which traps catch most students here?
- Using the first term as the constant, writing 4n + 5 when the rule is 4n + 1.
- Trusting a rule after one check, instead of testing a later term.
- Adding when the pattern multiplies, because the differences were never compared.
- Using the whole second difference, so the n² coefficient comes out twice too big.
- Substituting a value where n belongs, which mixes up position and value.
Each lesson shows one of these slips in full and then corrects it.
How should you use the practice set?
Attempt each question on paper before opening the answer. Write the working you would show in an exam, including your check. The sequence and series laboratory lets you generate terms from a rule and compare two candidate rules, and the non-calculator working trainer helps with the arithmetic.
When a question goes wrong, use the routing list at the end of the practice set and return to the lesson it names. The mistake log and retest queue helps you retry a fresh question a few days later.
If you want teaching beyond self-study, online one-to-one Mathematics tuition means an experienced teacher looks at your written working and finds which habit sits behind the error.