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Nth Term Sequences: Free Printable Worksheet for Year 8

  • ForYear 8
  • StageKey Stage 3
  • CostFree
  • FormatPrintable PDF
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Nth Term Sequences is a free printable maths worksheet for Year 8 (ages 12 to 13), in Key Stage 3. Pupils fill the gaps and find the rule for linear sequences. They find and use the nth term, which links to the KS3 aim of generating terms from a position to term rule. It needs only a pencil, and page 2 gives the grown-up notes, help and a stretch idea.

It comes from the Mini Minds Hub, the free library of printables from Jaques of London, the British games and toy company that has been making games for families since 1795. Print both pages, give page 1 to your pupil and keep page 2 for yourself. Black and white printing works well.

What is on the sheet

Pupils work with linear sequences. They fill the gaps and find the rule. They look at how a sequence changes from one term to the next, then use that change to build the nth term. They can also use a rule to find terms at a given position. Page 1 is the activity and needs only a pencil. Encourage pupils to show their working beside each answer.

A worked example

Find the nth term of the sequence 11, 18, 25, 32, ...

This is a separate sequence, used only to show the method. Two ways of working are shown below.

Method 1: use the difference
  1. The terms go up by 7 each time, so the rule starts with 7n.
  2. Test 7n at n = 1. 7 times 1 is 7. The first term is 11, so we need to add 4.
  3. The nth term is 7n + 4.
Method 2: work back to term zero
  1. The difference is 7. Go back one step from the first term: 11 take away 7 is 4.
  2. This is the term before the first one. It is the number added to 7n.
  3. Check with n = 4. 7 times 4 is 28, and 28 + 4 is 32. This matches the fourth term.
The nth term is 7n + 4

How the nth term is built

Step 1: find the difference

Subtract each term from the next one. In a linear sequence the change is the same every time. This number goes in front of n.

Step 2: write the n part

If the sequence goes up by 5, begin with 5n. If it goes down by 5, begin with a negative 5n. This gives the multiples that the sequence is built on.

Step 3: adjust

Compare the first term with the first value of your n part. Work out what must be added or taken away to match. That number goes at the end of the rule.

Step 4: check

Put n = 1, n = 2 and n = 3 into your rule. If it gives the terms you started with, the rule works.

What is an nth term?

A sequence is a list of numbers that follow a rule. Each number is called a term. In algebra, n stands for the position of a term. The first term has n = 1, the second has n = 2, and so on. An nth term rule is a short way to find any term from its position, without writing out all the ones before it.

A linear sequence changes by the same amount each time. That is why its nth term has the form of a multiple of n, plus or minus a number. Spotting patterns like this is the start of algebraic thinking. It also prepares pupils for straight line graphs, where the same constant change appears as a gradient.

Skills and curriculum link

  • Linear sequences
  • Finding the nth term
  • Algebraic notation
  • Term to term patterns
  • Checking by substitution

Ideas that help with nth term

  • Position and termThe position says where a term sits. The term is the number itself. Keep these two ideas apart and the rule is easier to follow.
  • The common differenceThe change between terms tells you the multiple of n. A bigger change means the sequence grows faster.
  • SubstitutionPutting a position into the rule gives the term. It is a quick way to test any rule you have written.
  • Term to term and position to termA term to term rule tells you how to get from one term to the next. A position to term rule, the nth term, lets you jump straight to any term.

Key words

  • TermOne number in a sequence.
  • SequenceA list of numbers that follow a rule.
  • Linear sequenceA sequence that goes up or down by the same amount each time.
  • Common differenceThe amount a linear sequence changes by each step.
  • nth termA rule that uses n for the position and gives any term in the sequence.
  • SubstituteReplace a letter with a number so you can work out the value.

How to use it

  1. Print both pages. Page 1 is for the pupil. Keep page 2, which has notes for you.
  2. Work through the first gap together. Ask the pupil to say out loud what is changing from one term to the next.
  3. Let them finish the rest alone. They can use a table with n above and the term below if it helps.
  4. Check against the notes on page 2. Ask, "How did you find the rule?" The method matters as much as the answer.
  5. Come back to it another day. Repeating a familiar activity builds confidence.

Help, stretch and game ideas

If it feels too hard

Draw the sequence as a table, with n above and the term below. Write the positions 1, 2, 3 and so on across the top. Look at how much the terms change each step. That change is the number in front of n. Do the first gap together.

To stretch confident children

Write your own linear sequence and swap it with a partner, who must find the nth term. Then check the rule by using it to rebuild your sequence. For more challenge, write one that goes down each time, or find a term far along it.

Make it a game

Play rule detective. One player secretly picks a rule such as 6n + 1 and writes the first four terms. The other player finds the rule. Swap roles and score a point for each rule found. Or time pupils and count the nth terms found.

Mistakes to watch for

Watch forA pupil writes n plus the difference, such as n + 4 for a sequence that goes up in fours.

Try thisThe difference is the number that multiplies n. Test the rule with n = 1 and n = 2 and see that it does not match.

Watch forA pupil gets the 'n' part right but forgets the adjustment at the end.

Try thisSubstitute n = 1 into the n part and compare it with the first term. The gap between them is what must be added or taken away.

Watch forWith a sequence that goes down, a pupil writes the difference as positive.

Try thisCheck the direction. If the terms are getting smaller, the number in front of n is negative. Test the rule with n = 2.

Talk about it

  • How can you tell whether a sequence is linear?
  • What does n stand for in an nth term rule?
  • Why does the change between terms become the number in front of n?
  • How could you check a rule without writing out the whole sequence?
  • What is the difference between a term to term rule and an nth term rule?
  • Can you make up a sequence that has a very large nth term but starts small?

Take it off the page

Matchstick patterns

Build a pattern of shapes with matchsticks or paper clips. Count how many are used in each shape and write the sequence. Then find a rule for any shape in the pattern.

Walking steps

Count your steps along a path, noting the total at each stop. Look for a rule that links stop number to steps taken.

Saving plan

Imagine saving the same amount every week from a starting sum. Write the total after each week as a sequence and find its nth term.

Who it is for

Parents

Sit beside your child for the first gap, then let them lead. Short sessions work better than long ones. If they get stuck, ask them to draw a table with n above and the term below.

Teachers

Use it as a starter, a plenary, an early finisher task or homework. It needs no preparation beyond printing. Page 2 gives support and stretch ideas for mixed ability groups.

Home educators

Print it as part of a wider maths plan. It pairs well with other algebra work and with everyday talk about patterns. You can print it again for revision or for brothers and sisters.

Where this fits

Before this, pupils meet number patterns, find the next term and describe term to term rules in words. They also need to be at ease with using letters to stand for numbers, and with simple substitution.

After this, pupils can move on to sequences that are not linear, such as those with changing differences. Nth term work also supports straight line graphs and forming and solving equations.

Questions parents and teachers ask

What age is Nth Term Sequences for?

It is written for Year 8 (ages 12 to 13), in Key Stage 3. Younger pupils who are confident with number patterns may try it with support, and older pupils can use it for revision.

What will children practise?

Pupils find and use the nth term of a linear sequence, linking to the KS3 aim of generating terms from a position to term rule. They also practise spotting a constant difference and checking a rule.

How do I use it?

Work through the first gap together, then let pupils finish alone. Check against the notes on page 2, and talk about the method used. Come back to it another day to build confidence.

What if it feels too hard?

Draw the sequence as a table with n above and the term below. Look at the change between terms and use it to start the rule. Page 2 has more help for children who find it hard.

Is it free?

Yes. It is free to print for home and classroom use. You can print it again whenever you need it, for example for revision a few weeks later.

How long does it take?

It depends on the pupil. The sheet gives no set time, so let them work at their own pace. Page 2 has help for children who find it hard and a stretch idea for confident pupils.

Do I need to print in colour?

No. It prints well in black and white on plain A4 paper. It is also a screen free activity that needs only paper and a pencil.

Preview the full PDF

Made by Jaques of London, making games for families since 1795. A printable supports classroom learning and does not replace a teacher.

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