Year 6 Arithmetic (SATs): Linear number sequences
Part of the Year 6 Arithmetic (SATs) algebra and formulae chapter. Chapter overview · whole chapter on one page
Linear number sequences
Rule
A linear sequence goes up or down by the same amount each time. That amount is the term-to-term rule, which you can write in words: "add 3" or "subtract 4".
To find the 10th term of 5, 11, 17, ..., add the step 9 times: 5 + 9 × 6 = 59.
Key ideas
- Find the step between terms: add or subtract the same amount each time.
- The term-to-term rule says how to get from one term to the next.
- To find a far term, use the step, not a long list.
Method. Find the step, then add it on.
- Find the step. 7 - 4 = 3, and 10 - 7 = 3.step = 3
- Add the step. 13 + 3 = 16 and 16 + 3 = 19.16, 19
- Write the rule. The term-to-term rule is add 3.add 3
- Write the answer. 16 and 19
Check. The rule is add 3. ✓
Method. Find the step, then add or subtract it.
- Find the step. 17 - 20 = -3, so each term goes down by 3.step = -3
- Continue the list. 11 - 3 = 8, then 8 - 3 = 5.8, 5
- Read the 6th term. The 6th term is 5.6th term = 5
- Write the answer. subtract 3; 5
Check. 20, 17, 14, 11, 8, 5: the 6th term is 5. ✓
Method. Find the step, then add it 9 times to the first term.
- Find the step. 11 - 5 = 6, so the step is 6.step = 6
- Count the steps. The 10th term is 9 steps after the 1st term.9 steps
- Work it out. 5 + 9 × 6 = 5 + 54 = 59.59
- Write the answer. 59
Check. The 9th step gives 5 + 54 = 59. ✓
Exercise 6C Practice: linear number sequences
Non-calculatorMethod. Find the step, then add it to the last term.
- Find the step. Each term changes by 5.step = 5
- Add the step. 17 + 5 = 22.22
- Write the answer. 22
Check. The sequence goes up by 5 each time. ✓
Method. Find the step, then add it to the last term.
- Find the step. Each term changes by -4.step = -4
- Add the step. 18 - 4 = 14.14
- Write the answer. 14
Check. The sequence goes down by 4 each time. ✓
Method. Find the step, then add it to the last term.
- Find the step. Each term changes by 3.step = 3
- Add the step. 16 + 3 = 19.19
- Write the answer. 19
Check. The sequence goes up by 3 each time. ✓
Method. Find the step, then add it to the last term.
- Find the step. Each term changes by -6.step = -6
- Add the step. 32 - 6 = 26.26
- Write the answer. 26
Check. The sequence goes down by 6 each time. ✓
Method. Subtract one term from the next to find the step.
- Find the step. 9 - 6 = 3.3
- Write it in words. Each term is 3 more than the one before.add 3
- Write the answer. add 3
Check. 12 - 9 = 3 as well. ✓
Method. The step is 4, so find the term between 8 and 16.
- Find the step. 20 - 16 = 4.4
- Find the missing term. 8 + 4 = 12, and 12 + 4 = 16.12
- Write the answer. 12
Check. 12 + 4 = 16. ✓
Method. Add the step of 4 nine times to the first term.
- Find the step. 7 - 3 = 4.4
- Add nine steps. 3 + 9 × 4 = 3 + 36.39
- Write the answer. 39
Check. 3 + 36 = 39. ✓
Method. Each term goes up by 2. The 8th term is 7 steps on from the first.
- Find the step. 3 - 1 = 2.2
- Work it out. 1 + 7 × 2 = 15.15
- Write the answer. 15
Check. Check: 1, 3, 5, 7, 9, 11, 13, 15 has 8 terms. ✓
Method. Count the steps from 15 down to 3.
- Find the difference. 15 - 3 = 12.12
- Count the steps. 12 ÷ 4 = 3 steps, so 3 is the 4th term.3 steps, 4th term
- Write the answer. 4th term
Check. 15, 11, 7, 3 are the first four terms. ✓
Method. List the terms of each sequence, then find the numbers in both.
- List the first sequence. 4, 9, 14, 19, 24, 29, 34, 39 ...4, 9, 14, 19, 24, 29, 34, 39
- List the second sequence. 3, 9, 15, 21, 27, 33, 39, 45 ...3, 9, 15, 21, 27, 33, 39, 45
- Find the numbers in both. 9 and 39 appear in both lists.9, 39
- Write the answer. 9 and 39
Check. Both lists contain 9 and 39. ✓
Method. The terms go up by 5, so check whether 50 is 3 plus a multiple of 5.
- Find the difference. 50 - 3 = 47.47
- Check the step. 47 is not a multiple of 5, so 50 is not a term.47 ÷ 5 is not whole
- Write the answer. No
Check. 3, 8, 13, ..., 48, 53: 50 is missed. ✓
Method. Subtract 10 eleven times from 100.
- Find the step. 90 - 100 = -10.-10
- Work it out. 100 - 11 × 10 = 100 - 110 = -10.-10
- Write the answer. -10
Check. The sequence goes below zero after the 10th term. ✓
Method. A linear sequence has the same step each time.
- Find the steps. 4 - 2 = 2 and 8 - 4 = 4.2 and 4
- Compare. The steps are different, so it is not linear.not linear
- Write the answer. It is not linear because the step changes: 4 - 2 = 2 but 8 - 4 = 4, so the terms do not go up by the same amount.
Check. The steps 2 and 4 are not equal. ✓