Study case: finding the nth term of a linear sequence In this Year 8 Sequences nth Term Problem Tutor, I use one study case to help students understand how a sequence rule is built. The problem is: find the nth term of the sequence 5, 8, 11, 14, 17. The answer is 3n + 2. I like this example because it clearly shows the link between common difference, position number and term value. Before solving, I ask the student to find the common difference. The sequence increases by 3 each time. That tells us the nth term will start with 3n. However, 3n gives the sequence 3, 6, 9, 12, 15. Our sequence is 5, 8, 11, 14, 17. Each term is 2 more than the 3n sequence, so the nth term is 3n + 2. Method 1 is the comparison method. I write the position numbers: n = 1, 2, 3, 4, 5. Then I write 3n: 3, 6, 9, 12, 15. Comparing this with 5, 8, 11, 14, 17 shows that we need to add 2 each time. Therefore the rule is 3n + 2. This method is reliable because students can see exactly where the plus 2 comes from. Method 2 is the first term adjustment method. The common difference is 3, so the rule begins 3n. When n = 1, 3n gives 3. The actual first term is 5. To get from 3 to 5, add 2. Therefore the nth term is 3n + 2. This is a quick method once students are confident with the idea. Method 3 is the checking method. I test the rule with several positions. When n = 1, 3(1) + 2 = 5. When n = 2, 3(2) + 2 = 8. When n = 5, 3(5) + 2 = 17. The rule works for the listed terms, so it is correct. I teach students not to stop after finding a rule; they should always test it. After the methods, I compare them. The comparison method builds understanding. The first term adjustment method is fast. The checking method protects against mistakes. I want students to understand that the coefficient of n comes from the common difference. This is one of the most important ideas in linear sequences. Common mistakes include using the first term as the coefficient, writing 5n + 3, forgetting to compare with the 3n sequence, and not checking the answer. I use these mistakes as teaching clues. If a student writes 5n, I ask them to look at how much the terms increase by each time. The common difference, not the first term, controls the multiplier. This topic is important for KS3 because sequences prepare students for algebra, straight-line graphs and functions. The rule 3n + 2 is similar to y = 3x + 2, so sequence work helps students understand gradients and intercepts later. I often point out this connection for students who are ready. A useful extension is to find the nth term of 7, 11, 15, 19, where the common difference is 4. Another extension is to find a term far into the sequence, such as the 50th term, using the rule. This shows why nth term rules are useful: they let us find terms without listing every step. In one-to-one tutoring, I would finish by asking the student to create a sequence with nth term 4n - 1 and list the first five terms. Reversing the process helps confirm understanding. When a student can move from sequence to rule and from rule to sequence, the topic becomes much more secure. For support with this topic, students can book one-to-one tutor services: https://www.mastermathstutoring.co.uk/services. For the wider course page, visit KS3 Maths tutoring: https://www.mastermathstutoring.co.uk/ks3. To ask about lessons or availability, use contact MasterMaths Tutoring: https://www.mastermathstutoring.co.uk/contact-me.