Free printable number sequence chart
Grades 4–8Terms, positions, and the two kinds of rule — the one that gets you the next term and the one that gets you any term at all. Print it for the wall, or jump straight to the hundredth term of the sequences below.
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Two rules, and why the second one matters
Every sequence here shows its terms with their positions above them, and both rules underneath. Change the term number and it jumps straight there without counting.
That jump is the whole reason the position-to-term rule exists. Finding the 100th term of a sequence by adding seven ninety-nine times is possible and miserable. Knowing the rule is 7n + 2 makes it one multiplication.
Try the square and triangular numbers too. Their term-to-term rules are awkward — the gaps keep changing — but their position rules are simple: n × n, and n × (n + 1) ÷ 2. That is exactly the situation the position rule was invented for.
| Position | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Term | 5 | 7 | 9 | 11 | 13 |
- Term-to-term rule
- Add 2 each time
- Position-to-term rule
- 2n + 3
- Term 20
- 43
The term-to-term rule gets you the next one. The position-to-term rule gets you any one, without counting.
How to use this sequences chart at home
Ask for the 100th term early. It is the question that makes the position rule necessary — nobody wants to add seven ninety-nine times, and feeling that inconvenience is what motivates the algebra.
Write the positions above the terms, always. Sequences are usually presented as a bare list, and half the difficulty vanishes once 1, 2, 3, 4, 5 is written above them. The relationship the nth term describes is between those two rows, and it is invisible with only one.
Then check a rule rather than trusting it. If a child says the rule is 7n + 2, put in n = 3 and see whether 23 comes out. Substituting to check is the same skill they will need constantly in algebra.
Why the nth term is worth the trouble
The term-to-term rule is what children find first and what they prefer, because it is visible in the gaps. It also has a real limitation: it can only ever tell you the next one.
The position rule answers questions the other cannot. What is the 100th term? Is 500 in this sequence? Which position does 86 sit in? Each of those is a single calculation with the position rule and an afternoon of counting without it.
That is genuinely the start of algebra. The n in 7n + 2 is a letter standing for a position, and every expression a child meets later works the same way — a rule written once that covers infinitely many cases.
Finding the rule for an adding sequence
Take the common difference first. If the sequence goes up in 7s, the rule starts with 7n, because the seven times table is the closest thing to it.
Then line them up. Write 7, 14, 21, 28 underneath the actual sequence 9, 16, 23, 30. Every term is 2 bigger, so the rule is 7n + 2. The adjustment is always the same number, which is why this method works.
The common slip is using the first term as the adjustment. For 9, 16, 23 the rule is not 7n + 9. Comparing against the times table rather than guessing keeps it right, and substituting a value checks it in five seconds.
Where sequences usually go wrong
Confusing the term with its position. The 4th term of 9, 16, 23, 30 is 30, not 4. Writing the positions above prevents almost all of these.
Using the first term as the constant in the nth term rule. Compare with the times table instead.
Assuming every sequence adds. Square and triangular numbers do not, and hunting for a fixed gap in them leads nowhere.
Stating a rule without checking it. Substituting one position takes seconds and catches most errors.
After the chart goes up
A chart explains the idea but gives a child nothing to do. These do the rest.
- number sense at Mathify Kids — every topic in this area, sorted by grade.
- skip counting worksheets — printable practice.
- order of operations worksheets — the next step once this is solid.
- exponents worksheets — more practice in the same area.
Everything at this level: Grade 4 math · Grade 5 math · Grade 6 math · Grade 7 math · Grade 8 math
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Questions parents ask
What is a sequence?
A list of numbers in order, where each one has a position. The numbers themselves are called terms, and the first term sits at position 1. Sequences are patterns with the positions written down, which is what makes the second kind of rule possible.
What is the difference between the two rules?
The term-to-term rule tells you how to get from one term to the next — add 7. The position-to-term rule tells you how to get any term directly from its position — 7n + 2. The first is easier to spot, the second is far more useful.
How do you find the nth term of a sequence?
For a sequence that adds the same amount each time, the multiplier is that amount. So a sequence going up in 7s starts with 7n. Then compare 7n with your actual sequence and adjust: if 7n gives 7, 14, 21 and your sequence is 9, 16, 23, you add 2, giving 7n + 2.
What are square numbers?
The numbers you get by multiplying a whole number by itself: 1, 4, 9, 16, 25. They are called square because that many counters can be arranged into a perfect square. The nth term is simply n × n.
What are triangular numbers?
1, 3, 6, 10, 15 — the running totals of 1, then 1+2, then 1+2+3. They stack into triangles, which is where the name comes from. Two identical triangles fit together into a rectangle, which is why the rule is n × (n + 1) ÷ 2.
What grade is this sequences chart for?
Grades 4 to 8. Continuing sequences and describing term-to-term rules arrives around Grade 4, square and triangular numbers by Grade 5 or 6, and the nth term in Grades 6 to 8 as algebra begins.
Is this sequences chart free to print?
Yes. Open it full size and print as many copies as you need for home or a classroom, with no account and no watermark. Reselling it or re-hosting the file elsewhere is not allowed — link to this page instead.