Year 6: Linear number sequences: the term-to-term rule
Part of the Year 6 algebra chapter. Chapter overview · whole chapter on one page
Linear number sequences: the term-to-term rule
A sequence is a list of numbers that follow a rule. Each number in the list is a term.
In a linear sequence, the same amount is added each time, or the same amount is taken away each time. The rule that says how to get from one term to the next is the term-to-term rule. For example, 5, 9, 13, 17 goes up by 4 each time, so the rule is add 4.
Look at the dot patterns below. Each pattern has 2 more dots than the one before it, so the number of dots goes up in a linear sequence.
Key ideas
- Find the difference between two terms next to each other.
- If the difference is always the same, the sequence is linear, and the difference is the rule.
- To go forwards, add (or take away) the difference. To go backwards, do the opposite.
- A sequence that goes up by 4 and a sequence that goes down by 4 are both linear.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. The rule is to add 4 each time. Use it to find the next terms.
- Find the difference between two terms next to each other. If the difference is always the same, the sequence is linear.9 - 5 = 4
- Check the other pairs of terms. They must have the same difference, or the rule is not linear.9 - 5 = 4, 13 - 9 = 4, 17 - 13 = 4
- Apply the rule: add 4. This gives the next term.17 + 4 = 21
- Do the same again for the term after that. The same rule works every time.21 + 4 = 25
- Write the answer. 21, 25
Check. Start from the last term: 17 + 4 = 21. ✓
Watch out. Check that the difference is the same for every pair of neighbouring terms before you use it.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. Find the difference and say whether the terms go up or down.
- Find the difference between two terms next to each other. This is the amount we add or take away.36 - 40 = -4
- Check that every pair has the same difference. Then the rule is the same all the way along.4, 4, 4 (sizes of the jumps)
- Write the rule. Say whether we add or subtract, and by how much.subtract 4
Check. Start at 40 and apply the rule: 40 - 4 = 36. ✓
Watch out. Say "add 4" or "subtract 4", not "plus 4" or "4 more" with no operation.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. Work out how far the number is from the first term. If that is a whole number of jumps, the number is in the sequence.
- Take away the first term from 38. We want to know how far the term is from the start.38 - 3 = 35
- Divide by the difference, 5. The sequence goes up by the same amount each time.35 ÷ 5 = 7
- It is a whole number that is 0 or more, so the number is in the sequence. The position is one more than the number of jumps.position = 7 + 1 = 8, so it is term number 8
- Write the answer. Yes
Check. Check: 3 + 7 × 5 = 38. ✓
Watch out. The number must be exactly one of the terms. A number between two terms is not in the sequence.
Watch out
A linear sequence must have the same difference all the way along. For 1, 4, 9, 16 the differences are 3, 5, 7, so it is not linear.
Exam tip
Say the rule in words: “add 5” or “subtract 4”. Then check it with the first two terms.
Exercise 8C Terms, rules and patterns
Non-calculatorKey words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. The rule is to add 5 each time. Use it to find the next terms.
- Find the difference between two terms next to each other. If the difference is always the same, the sequence is linear.11 - 6 = 5
- Check the other pairs of terms. They must have the same difference, or the rule is not linear.11 - 6 = 5, 16 - 11 = 5, 21 - 16 = 5
- Apply the rule: add 5. This gives the next term.21 + 5 = 26
- Write the answer. 26
Check. Start from the last term: 21 + 5 = 26. ✓
Watch out. Check that the difference is the same for every pair of neighbouring terms before you use it.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. Find the difference and say whether the terms go up or down.
- Find the difference between two terms next to each other. This is the amount we add or take away.7 - 2 = 5
- Check that every pair has the same difference. Then the rule is the same all the way along.5, 5, 5 (sizes of the jumps)
- Write the rule. Say whether we add or subtract, and by how much.add 5
Check. Start at 2 and apply the rule: 2 + 5 = 7. ✓
Watch out. Say "add 4" or "subtract 4", not "plus 4" or "4 more" with no operation.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. The missing term is halfway between its two neighbours, so add them and halve.
- Look at the gap: it is one jump from each side. So the missing term is halfway between the two terms.30, __, 22
- Add the two terms. The two jumps together make the whole gap.30 + 22 = 52
- Halve the answer. This gives the middle term.52 ÷ 2 = 26
- Write the answer. 26
Check. 26 - 30 = -4 and 22 - 26 = -4: both jumps are the same. ✓
Watch out. Check that both jumps are the same size. If they are not, the sequence is not linear.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. The missing term is halfway between its two neighbours, so add them and halve.
- Look at the gap: it is one jump from each side. So the missing term is halfway between the two terms.12, __, 20
- Add the two terms. The two jumps together make the whole gap.12 + 20 = 32
- Halve the answer. This gives the middle term.32 ÷ 2 = 16
- Write the answer. 16
Check. 16 - 12 = 4 and 20 - 16 = 4: both jumps are the same. ✓
Watch out. Check that both jumps are the same size. If they are not, the sequence is not linear.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. Find the difference and say whether the terms go up or down.
- Find the difference between two terms next to each other. This is the amount we add or take away.45 - 50 = -5
- Check that every pair has the same difference. Then the rule is the same all the way along.5, 5, 5 (sizes of the jumps)
- Write the rule. Say whether we add or subtract, and by how much.subtract 5
Check. Start at 50 and apply the rule: 50 - 5 = 45. ✓
Watch out. Say "add 4" or "subtract 4", not "plus 4" or "4 more" with no operation.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. The rule is to subtract 8 each time. Use it to find the next terms.
- Find the difference between two terms next to each other. If the difference is always the same, the sequence is linear.92 - 100 = -8
- Check the other pairs of terms. They must have the same difference, or the rule is not linear.92 - 100 = -8, 84 - 92 = -8
- Apply the rule: subtract 8. This gives the next term.84 - 8 = 76
- Write the answer. 76
Check. Start from the last term: 84 - 8 = 76. ✓
Watch out. Check that the difference is the same for every pair of neighbouring terms before you use it.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. The rule is to add 2 each time. Use it to find the next terms.
- Find the difference between two terms next to each other. If the difference is always the same, the sequence is linear.6 - 4 = 2
- Check the other pairs of terms. They must have the same difference, or the rule is not linear.6 - 4 = 2, 8 - 6 = 2, 10 - 8 = 2
- Apply the rule: add 2. This gives the next term.10 + 2 = 12
- Write the answer. 12
Check. Start from the last term: 10 + 2 = 12. ✓
Watch out. Check that the difference is the same for every pair of neighbouring terms before you use it.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. Find the difference between each pair of neighbouring terms. If it changes, the sequence is not linear.
- Find the differences. Subtract each term from the next one.4 - 1 = 3, 9 - 4 = 5, 16 - 9 = 7, 25 - 16 = 9
- Compare the differences. A linear sequence has the same difference every time.3, 5, 7, 9 are not all the same
- Write the answer. No
Check. Adding 3 gives 4, but adding 3 again gives 7, not 9. So the rule is not linear. ✓
Watch out. The squares 1, 4, 9, 16, 25 go up by more and more each time.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. Check that every difference is the same.
- Find the differences. Each pair of neighbouring terms is compared.5 - 2 = 3, 8 - 5 = 3, 11 - 8 = 3
- Compare the differences. They are all 3, so the sequence is linear and the rule is add 3.the differences are all 3
- Write the answer. Yes
Check. 2 + 3 = 5, 5 + 3 = 8, 8 + 3 = 11. ✓
Watch out. All the differences must be the same.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. Work backwards: take the difference away from the term you know.
- Find the difference. The sequence goes up by 5 each time.17 - 12 = 5
- Go back one term. Take the difference away from 12.12 - 5 = 7
- Write the answer. 7
Check. 7 + 5 = 12, and 12 + 5 = 17. ✓
Watch out. Going backwards means doing the opposite operation.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. Start at 3 and add 6 each time.
- Start with the first term. The first term is given.3
- Add 6 to get the next term. Apply the rule once.3 + 6 = 9
- Add 6 again. Keep applying the same rule.9 + 6 = 15, then 15 + 6 = 21, then 21 + 6 = 27
- Write the answer. 3, 9, 15, 21, 27
Check. Each term is 6 more than the one before. ✓
Watch out. Start from the first term, not from zero.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. Work out how far the number is from the first term. If that is a whole number of jumps, the number is in the sequence.
- Take away the first term from 100. We want to know how far the term is from the start.100 - 4 = 96
- Divide by the difference, 4. The sequence goes up by the same amount each time.96 ÷ 4 = 24
- It is a whole number that is 0 or more, so the number is in the sequence. The position is one more than the number of jumps.position = 24 + 1 = 25, so it is term number 25
- Write the answer. Yes
Check. Check: 4 + 24 × 4 = 100. ✓
Watch out. The number must be exactly one of the terms. A number between two terms is not in the sequence.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. Compare the first terms and the rule. Both sequences go up by 7.
- Write out the first terms of each sequence. This shows the pattern clearly.A: 2, 9, 16, 23, ... B: 5, 12, 19, 26, ...
- Look at the remainders when each term is divided by 7. Every term of A leaves 2 when divided by 7. Every term of B leaves 5.2 and 5 are different, so the two lists can never match
- Write the answer. No
Check. Sequence A never shares a term with sequence B, because both go up by 7 but start at different remainders. ✓
Watch out. Two sequences with the same difference can still never meet if they start in different places.