# Arithmetic Sequences and the nth Term Explained Author: Rahul Sijwali Author URL: https://www.codeyoung.com/blog/author/rahul-sijwali Published: 2026-10-05 Category: Math For Kids Category URL: https://www.codeyoung.com/blog/category/math-for-kids Meta Title: Arithmetic Sequences and the nth Term Explained Meta Description: Arithmetic sequences are easy until the nth term arrives. Why the formula has n minus 1 in it, and the jump-counting fix for the mistake every child makes. Tags: Math for kids, Learn Math, Math Tips for Students, Arithmetic Sequences Tag URLs: Math for kids (https://www.codeyoung.com/blog/tag/math-for-kids), Learn Math (https://www.codeyoung.com/blog/tag/learn-math), Math Tips for Students (https://www.codeyoung.com/blog/tag/math-tips-for-students), Arithmetic Sequences (https://www.codeyoung.com/blog/tag/arithmetic-sequences) URL: https://www.codeyoung.com/blog/arithmetic-sequences-nth-term Ask a ten-year-old for the next number in 4, 7, 10, 13 and they will tell you instantly. Ask the same child for the 50th number and most will start writing out the list. Ask a thirteen-year-old for the nth term and a large share will write 4 + 3n, which is wrong, and will be unable to say why it feels right. That is not carelessness. **Arithmetic sequences** are the first place in school mathematics where a child has to stop asking "what comes next" and start asking "what sits at position n", and nothing in the two years before that has prepared them for the switch. The fix is one idea, and it is not a formula: **count the jumps, not the terms.** ## What this post covers - What an arithmetic sequence is, and what the **common difference** does. - Why **term-to-term** and **position-to-term** rules answer different questions. - Where the **n minus 1** comes from, explained without algebra. - The mistake almost every child makes first, and the check that catches it in five seconds. - When schools actually teach this, so you know whether your child is behind or early. ## What is an arithmetic sequence? It is a list of numbers where you add the same amount to get from each term to the next. That fixed amount is the **common difference**. In 4, 7, 10, 13, 16 the common difference is 3. In 20, 17, 14, 11 it is negative 3. In 3, 3.5, 4, 4.5 it is 0.5. All three are arithmetic sequences, because in each case one single number does all the work. Two pieces of vocabulary are worth settling before anything else, because they cause more trouble than the mathematics does. A **term** is one of the numbers in the list, not a word. And the **first term** is at position 1, not position 0. Children who are quietly unsure which of those is meant will guess, and the guess is invisible in their written work. Mathematical words that mean something different in ordinary English are a reliable source of lost marks, which we have written about separately in [the maths words that trip children up](https://www.codeyoung.com/blog/maths-words-that-trip-children-up). ## Term-to-term and position-to-term: two different questions This is the distinction the whole topic turns on, and most explanations skip straight past it to the formula. A **term-to-term rule** answers: how do I get from one term to the next? For 4, 7, 10, 13 the answer is "add 3". It is a set of instructions you follow repeatedly. A **position-to-term rule** answers: how do I get the term at position n directly, without walking through the ones before it? For the same sequence the answer is 3n + 1. It is a machine you feed a position into. The [national curriculum for mathematics](https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study) teaches those two things years apart, and the gap explains almost everything about why the nth term feels so sudden. StageWhat is expected**Year 5**"recognise and describe linear number sequences, including those involving fractions and decimals, and find the term-to-term rule in words"**Year 6**"generate and describe linear number sequences"**Key stage 3**"generate terms of a sequence from either a term-to-term or a position-to-term rule" and "recognise arithmetic sequences and find the nth term" Look at the Year 5 wording: the rule is to be found **in words**. For two full years a child is rewarded for saying "add a half" and is never once asked for a position rule. Then at key stage 3 the question changes shape, and the habit that earned them marks for two years now produces a wrong answer. That is a curriculum design fact, not a failing in the child, and it is worth saying out loud to a thirteen-year-old who thinks they have suddenly become bad at sequences. ## Where does the n minus 1 come from? From counting jumps. Take 4, 7, 10, 13, 16 and ask how many additions it takes to reach each term, starting from the first. PositionTermJumps of 3 from the start14027131024133516**4** The 5th term needs **four** jumps, not five. The first term is where you are standing, not somewhere you travelled to. Position 3 needs two jumps, position 10 needs nine, and position n needs n minus 1. So the 5th term is 4 + 3 × 4 = 16, which matches the list. And the general rule is the first term plus (n minus 1) lots of the common difference. Written out: 4 + 3(n − 1), which tidies up to 3n + 1. I would teach the jump table before the formula every time, and leave the formula until the child can already produce the right answer from the table. A child who has counted the jumps can rebuild the formula whenever they forget it. A child who memorised the formula and forgot it has nothing. That ordering, reasoning first and the formal method second, is the thing that makes sequences stick rather than being relearned every term. Not sure which level your child should start at? A free trial class with a Codeyoung teacher shows you exactly where they are and what they are ready for next, before you commit to anything. [Book a Free Trial →](https://book-a-demo.codeyoung.com?utm_source=blog&utm_medium=codeyoung&utm_campaign=arithmetic-sequences-nth-term) ## The mistake almost every child makes first They write first term plus n lots of the difference, instead of n minus 1 lots. For 4, 7, 10, 13, 16 that gives: **Wrong:** 4 + 3 × 5 = 19 for the 5th term. **Right:** 4 + 3 × 4 = 16 for the 5th term. The wrong answer is out by exactly one common difference, every single time. It lands on the 6th term rather than the 5th. Once a child knows that signature they can spot their own error without being told, which is far better than being corrected. There is a reason this particular slip is so stubborn. The child is doing precisely what two years of school taught them to do, which is to apply the adding rule, and applying it five times for the fifth term feels like consistency rather than error. Telling them to "remember the minus one" treats a conceptual mismatch as a memory problem. Showing them the jump table treats it as what it is. And the check takes five seconds. **Substitute n = 1 and see whether you get the first term.** For the correct rule 3n + 1, putting n = 1 gives 4, which is right. For the wrong instinct 3n + 4, putting n = 1 gives 7, which is the second term, and the error announces itself. Getting into the habit of sanity-checking an answer rather than trusting the arithmetic is worth more than any individual topic, something we have argued at length in [asking whether an answer is sensible](https://www.codeyoung.com/blog/is-this-answer-sensible). ![Infographic showing the sequence 4, 7, 10, 13, 16 with four arcs labelled plus 3 hopping between the five terms and one long arc beneath labelled four jumps, alongside the wrong calculation 4 plus 3 times 5 equals 19 against the correct 4 plus 3 times 4 equals 16, and a curriculum rail showing term-to-term in Year 5 and the nth term at key stage 3](https://prod.superblogcdn.com/site_cuid_clvc4016q001j13bhaleswmt1/images/arithmetic-sequences-nth-term-infographic-1791196340289-compressed.png)Four terms apart means four jumps, which is where the n minus 1 comes from. ## How to find the nth term from a sequence Four steps, and the fourth is not optional. 1. **Find the common difference.** Subtract any term from the one after it. For 7, 11, 15, 19 that is 4. 2. **Write down that number times n.** So 4n, which produces 4, 8, 12, 16. 3. **Compare with the real sequence and adjust.** The real one is 7, 11, 15, 19, which is 3 more each time. So the rule is 4n + 3. 4. **Check with n = 1.** 4 × 1 + 3 = 7, which is the first term. Correct. Step 3 is sometimes taught as "find the zeroth term", the value that would sit before the first one. For 7, 11, 15, 19 that imaginary term is 3, which is indeed the number you add. The shortcut is sound and children like it, but it is worth being clear that it is the same calculation written differently rather than a separate trick. A child who thinks it is a separate trick will apply it to a sequence where the differences are not constant and get an answer that looks plausible. Two more worked examples, because one is never enough: - **20, 17, 14, 11.** Common difference negative 3. Start with −3n, giving −3, −6, −9, −12. The real sequence is 23 more each time, so the rule is 23 − 3n. Check: 23 − 3 = 20. Correct. - **3, 3.5, 4, 4.5.** Common difference 0.5. Start with 0.5n, giving 0.5, 1, 1.5, 2. The real sequence is 2.5 more, so the rule is 0.5n + 2.5. Check: 0.5 + 2.5 = 3. Correct. That last one is taken straight from the Year 5 curriculum example, and it is a useful one to practise because children who are fluent with whole-number sequences often freeze when the difference is a fraction, even though not a single step of the method changes. ## Practice that builds the idea rather than the drill Twenty near-identical questions will produce a child who can find the nth term of sequences that look exactly like those twenty and nothing else. These five take longer and transfer further. 1. **Give them the rule and ask for the sequence.** "The rule is 5n − 2. Write the first four terms." Running the machine forwards is easier than building it and makes the structure visible. 2. **Ask for a term they cannot reach by counting.** The 100th term of 6, 11, 16. If they start listing, the question has done its job and you can ask whether there is a faster way. 3. **Ask whether a number is in the sequence.** Is 61 in 6, 11, 16? Solve 5n + 1 = 61, so n = 12. Is 63? That gives n = 12.4, which is not a position, so no. This is the first genuinely useful thing the formula does and it is usually left out. 4. **Mix in a sequence that is not arithmetic.** 2, 4, 8, 16. Ask for the common difference and let them discover there is not one. Children who have only ever been shown arithmetic sequences assume every list has a fixed difference. 5. **Have them build one for you.** "Write a sequence whose 4th term is 20." There are many answers, which is the point, and inventing one requires working the logic backwards. Question 3 is the one I would not skip. It converts the nth term from a thing you are asked for in exams into a tool that answers a question you could not otherwise answer, and that shift is what stops a topic feeling arbitrary. Children who stall partway through multi-step reasoning like this often have the same underlying issue described in [why two-step word problems stall](https://www.codeyoung.com/blog/two-step-word-problems-stall). ## Is my child behind? Probably not, and the curriculum table above is the thing to check against rather than what another child in the class can do. If your child is in Year 5 or 6 and can describe a sequence in words, they are exactly where they should be, and pushing the nth-term formula at them now will teach memorisation rather than understanding. If they are 12 or 13 and the nth term is not landing, start with the jump table and the n = 1 check rather than with more questions. A child with secure [number sense](https://www.codeyoung.com/blog/number-sense-for-kids-guide) normally picks the whole topic up in two or three sessions once the jump idea is in place. A child without it will struggle on the arithmetic rather than on the sequences, and more sequence practice will not help. Codeyoung runs 1:1 live online classes for children aged 6 to 17, with a teacher who adapts the pace to your child rather than a fixed syllabus. The first class is free, so you can see how they respond before deciding. [Book a Free Trial](https://book-a-demo.codeyoung.com?utm_source=blog&utm_medium=codeyoung&utm_campaign=arithmetic-sequences-nth-term) Our [online maths classes for kids](https://www.codeyoung.com/math/online-math-classes-for-kids) work this way throughout: the reasoning first, the formal method once the reasoning is secure. ## FAQs Q: What is an arithmetic sequence? A: A list of numbers where you add the same amount every time to get from one term to the next. In 4, 7, 10, 13 that amount is 3, and it is called the common difference. Subtracting a fixed amount counts too, which just means the common difference is negative. Q: Why does the nth term formula use n minus 1? A: Because you make one fewer jump than the term number. Getting to the 5th term from the 1st takes four additions, not five, since the first term is where you start rather than somewhere you jump to. That single idea is the whole reason for the minus one. Q: What is the difference between term-to-term and position-to-term? A: Term-to-term tells you how to get from one term to the next, such as add 3. Position-to-term tells you how to get any term straight from its position, such as 3n plus 1. The national curriculum teaches the first in Year 5 and the second at key stage 3. Q: How do you find the nth term of an arithmetic sequence? A: Find the common difference, multiply it by n, then adjust. For 4, 7, 10, 13 the difference is 3, so start with 3n, which gives 3, 6, 9, 12. Every value is one short, so the rule is 3n plus 1. Always check by substituting n equals 1. Q: When do children learn the nth term at school? A: At key stage 3, usually age 11 to 14. Before that, Year 5 asks children to find the term-to-term rule in words and Year 6 to generate and describe linear sequences. Degrees of formal algebra are not expected in primary school at all. Q: Can an arithmetic sequence go downwards? A: Yes. In 20, 17, 14, 11 the common difference is negative 3, so the nth term is 23 minus 3n. Children often find these easier once they see that nothing changes in the method, only the sign of the number they are adding each time. --- This blog is powered by Superblog. Visit https://superblog.ai to know more. ---