GCSE Maths: Sequences and term-to-term rules (Foundation)
Part of the GCSE Maths sequences and nth term chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Sequences and term-to-term rules
A sequence is a list of numbers, in a fixed order, that follow a rule. You meet sequences all the time. A bus leaves at 9:00, 9:20, 9:40, … A phone plan costs £10 in month 1, £14 in month 2, £18 in month 3, … Each number in the list is called a term. We say the first term, the second term, the third term, and so on. A sequence that gets bigger is increasing (ascending) and one that gets smaller is decreasing (descending).
The simplest way to describe a sequence is the term-to-term rule. This tells you what to do to one term to get the next term. In the sequence 3, 7, 11, 15, … you add 4 each time, so the term-to-term rule is “add 4”. The arrows below show it.
To find the rule, look at how each term changes into the next. Always try these checks in this order:
- Differences. Subtract each term from the one after it. If the answer is always the same, the rule is “add that number” (or “subtract” if the answers are negative).
- Ratios. If the differences are not the same, divide each term by the one before it. If the answer is always the same, the rule is “multiply by that number” (or “divide by” when the sequence shrinks).
- Changing differences. Sometimes the amount you add changes in a regular way, for example +1, +2, +3, …
Some rules have two steps, such as “multiply by 2 and then add 1”. Starting at 3 you get 3, then 3 × 2 + 1 = 7, then 7 × 2 + 1 = 15, and so on. When a rule is given to you, the question will also tell you the first term, because a rule alone cannot tell you where to start.
Finally, to continue a sequence, always apply the rule to the last term you have. Do not go back to the start each time.
Key ideas
- A sequence is an ordered list of numbers; each number is a term.
- The term-to-term rule says what to do to one term to get the next.
- Check the differences first (add or subtract), then the ratios (multiply or divide).
- Sometimes the difference itself changes by the same amount each time (+1, +2, +3, …).
- To continue a sequence, apply the rule to the last term.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Find the rule that takes you from one term to the next, then use it to find the next 2 terms.
- Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.9 − 5 = 4
13 − 9 = 4
17 − 13 = 4 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 4, so the rule is add 4.
- Use the rule to carry on the sequence. Apply the rule to the last term you have, then to the new term, and so on.17 + 4 = 21
21 + 4 = 25 - Write down the new terms. 21, 25
Check. Jump straight there: 5 + 5 × 4 = 25, and the last term we found is 25. ✓
Watch out. Work out the rule from the terms you are given, and use it on the last term. Do not start again from the first term.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Find the rule that takes you from one term to the next, then use it to find the next 2 terms.
- Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.44 − 50 = −6
38 − 44 = −6
32 − 38 = −6 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is −6, so the rule is subtract 6.
- Use the rule to carry on the sequence. Apply the rule to the last term you have, then to the new term, and so on.32 − 6 = 26
26 − 6 = 20 - Write down the new terms. 26, 20
Check. Jump straight there: 50 − 5 × 6 = 20, and the last term we found is 20. ✓
Watch out. Work out the rule from the terms you are given, and use it on the last term. Do not start again from the first term.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Find the rule that takes you from one term to the next, then use it to find the next 2 terms.
- Try the differences first. We always look for "add or subtract the same number" before anything else.6 − 3 = 3
12 − 6 = 6
24 − 12 = 12
The differences are not all the same, so it is not an add/subtract rule. - Now divide each term by the term before it. If we are multiplying by the same number each time, dividing a term by the one before gives that number.6 ÷ 3 = 2
12 ÷ 6 = 2
24 ÷ 12 = 2 - Look at the answers. They are all the same, so we multiply by that number each time.The rule is multiply by 2.
- Use the rule to carry on the sequence. Apply the rule to the last term you have, then to the new term, and so on.24 × 2 = 48
48 × 2 = 96 - Write down the new terms. 48, 96
Check. Look back along the whole list: 12, 24, 48, 96. Every term is the one before multiplied by 2. ✓
Watch out. Work out the rule from the terms you are given, and use it on the last term. Do not start again from the first term.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. A term-to-term rule tells you what to do to one term to get the next. Start with the first term and apply the rule again and again.
- Write down the first term. The first term is given, so it is the start of the list.Term 1 = 2
- Apply the rule "multiply by 2 and then add 1" to find each new term. Each term comes from the term before it. Use the answer you have just found as the next starting point.Term 2: 2 × 2 = 4, then 4 + 1 = 5
Term 3: 5 × 2 = 10, then 10 + 1 = 11
Term 4: 11 × 2 = 22, then 22 + 1 = 23
Term 5: 23 × 2 = 46, then 46 + 1 = 47 - Write the terms as a list. 2, 5, 11, 23, 47
Check. Test the rule on the last pair: 23 × 2, then + 1 gives 47. ✓
Watch out. Do the operations in the order the rule gives them. "Multiply by 2 then add 1" is not the same as "add 1 then multiply by 2".
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Find the rule that takes you from one term to the next, then use it to find the next 2 terms.
- Try the differences first. We always look for "add or subtract the same number" before anything else.2 − 1 = 1
4 − 2 = 2
7 − 4 = 3
11 − 7 = 4
The differences are not the same. - Look at how the differences change. Sometimes the amount added grows by the same amount each time.1, 2, 3, 4
The differences go up by 1 each time. - Write the rule. The differences increase by 1 each time: add 1, then add 2, then add 3, and so on.
- Use the rule to carry on the sequence. Apply the rule to the last term you have, then to the new term, and so on.11 + 5 = 16 (the next difference is 5)
16 + 6 = 22 (the next difference is 6) - Write down the new terms. 16, 22
Check. Look at the differences of the whole list: 1, 2, 3, 4, 5, 6. They keep increasing by 1. ✓
Watch out. Work out the rule from the terms you are given, and use it on the last term. Do not start again from the first term.
Watch out
The rule “add 3” does not tell you where the sequence starts: 2, 5, 8, … and 10, 13, 16, … both use it. And be careful not to describe 3, 6, 12, 24 as “add 3”: the first step adds 3 but the second adds 6. A rule must work for every pair of neighbouring terms.Exercise 11A Term-to-term rules
Non-calculatorKey words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Find the rule that takes you from one term to the next, then use it to find the next 3 terms.
- Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.4 − 2 = 2
6 − 4 = 2
8 − 6 = 2 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 2, so the rule is add 2.
- Use the rule to carry on the sequence. Apply the rule to the last term you have, then to the new term, and so on.8 + 2 = 10
10 + 2 = 12
12 + 2 = 14 - Write down the new terms. 10, 12, 14
Check. Jump straight there: 2 + 6 × 2 = 14, and the last term we found is 14. ✓
Watch out. Work out the rule from the terms you are given, and use it on the last term. Do not start again from the first term.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Find the rule that takes you from one term to the next, then use it to find the next 3 terms.
- Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.20 − 10 = 10
30 − 20 = 10
40 − 30 = 10 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 10, so the rule is add 10.
- Use the rule to carry on the sequence. Apply the rule to the last term you have, then to the new term, and so on.40 + 10 = 50
50 + 10 = 60
60 + 10 = 70 - Write down the new terms. 50, 60, 70
Check. Jump straight there: 10 + 6 × 10 = 70, and the last term we found is 70. ✓
Watch out. Work out the rule from the terms you are given, and use it on the last term. Do not start again from the first term.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Find the rule that takes you from one term to the next, then use it to find the next 3 terms.
- Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.3 − 1 = 2
5 − 3 = 2
7 − 5 = 2 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 2, so the rule is add 2.
- Use the rule to carry on the sequence. Apply the rule to the last term you have, then to the new term, and so on.7 + 2 = 9
9 + 2 = 11
11 + 2 = 13 - Write down the new terms. 9, 11, 13
Check. Jump straight there: 1 + 6 × 2 = 13, and the last term we found is 13. ✓
Watch out. Work out the rule from the terms you are given, and use it on the last term. Do not start again from the first term.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Find the rule that takes you from one term to the next, then use it to find the next 3 terms.
- Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.45 − 50 = −5
40 − 45 = −5
35 − 40 = −5 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is −5, so the rule is subtract 5.
- Use the rule to carry on the sequence. Apply the rule to the last term you have, then to the new term, and so on.35 − 5 = 30
30 − 5 = 25
25 − 5 = 20 - Write down the new terms. 30, 25, 20
Check. Jump straight there: 50 − 6 × 5 = 20, and the last term we found is 20. ✓
Watch out. Work out the rule from the terms you are given, and use it on the last term. Do not start again from the first term.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Find the rule that takes you from one term to the next, then use it to find the next 2 terms.
- Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.7 − 4 = 3
10 − 7 = 3
13 − 10 = 3 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 3, so the rule is add 3.
- Use the rule to carry on the sequence. Apply the rule to the last term you have, then to the new term, and so on.13 + 3 = 16
16 + 3 = 19 - Write down the new terms. 16, 19
Check. Jump straight there: 4 + 5 × 3 = 19, and the last term we found is 19. ✓
Watch out. Work out the rule from the terms you are given, and use it on the last term. Do not start again from the first term.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Find the rule that takes you from one term to the next, then use it to find the next 2 terms.
- Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.26 − 30 = −4
22 − 26 = −4
18 − 22 = −4 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is −4, so the rule is subtract 4.
- Use the rule to carry on the sequence. Apply the rule to the last term you have, then to the new term, and so on.18 − 4 = 14
14 − 4 = 10 - Write down the new terms. 14, 10
Check. Jump straight there: 30 − 5 × 4 = 10, and the last term we found is 10. ✓
Watch out. Work out the rule from the terms you are given, and use it on the last term. Do not start again from the first term.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Find the rule that takes you from one term to the next, then use it to find the next 2 terms.
- Try the differences first. We always look for "add or subtract the same number" before anything else.10 − 5 = 5
20 − 10 = 10
40 − 20 = 20
The differences are not all the same, so it is not an add/subtract rule. - Now divide each term by the term before it. If we are multiplying by the same number each time, dividing a term by the one before gives that number.10 ÷ 5 = 2
20 ÷ 10 = 2
40 ÷ 20 = 2 - Look at the answers. They are all the same, so we multiply by that number each time.The rule is multiply by 2.
- Use the rule to carry on the sequence. Apply the rule to the last term you have, then to the new term, and so on.40 × 2 = 80
80 × 2 = 160 - Write down the new terms. 80, 160
Check. Look back along the whole list: 20, 40, 80, 160. Every term is the one before multiplied by 2. ✓
Watch out. Work out the rule from the terms you are given, and use it on the last term. Do not start again from the first term.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Find the rule that takes you from one term to the next, then use it to find the next 2 terms.
- Try the differences first. We always look for "add or subtract the same number" before anything else.32 − 64 = −32
16 − 32 = −16
8 − 16 = −8
The differences are not all the same, so it is not an add/subtract rule. - Now divide each term by the term before it. If we are multiplying by the same number each time, dividing a term by the one before gives that number.32 ÷ 64 = 12
16 ÷ 32 = 12
8 ÷ 16 = 12 - Look at the answers. They are all the same, so we multiply by that number each time.The rule is divide by 2.
- Use the rule to carry on the sequence. Apply the rule to the last term you have, then to the new term, and so on.8 ÷ 2 = 4
4 ÷ 2 = 2 - Write down the new terms. 4, 2
Check. Look back along the whole list: 16, 8, 4, 2. Every term is the one before divided by 2. ✓
Watch out. Work out the rule from the terms you are given, and use it on the last term. Do not start again from the first term.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Find the rule that takes you from one term to the next, then use it to find the next 2 terms.
- Try the differences first. We always look for "add or subtract the same number" before anything else.2 − 1 = 1
4 − 2 = 2
7 − 4 = 3
11 − 7 = 4
The differences are not the same. - Look at how the differences change. Sometimes the amount added grows by the same amount each time.1, 2, 3, 4
The differences go up by 1 each time. - Write the rule. The differences increase by 1 each time: add 1, then add 2, then add 3, and so on.
- Use the rule to carry on the sequence. Apply the rule to the last term you have, then to the new term, and so on.11 + 5 = 16 (the next difference is 5)
16 + 6 = 22 (the next difference is 6) - Write down the new terms. 16, 22
Check. Look at the differences of the whole list: 1, 2, 3, 4, 5, 6. They keep increasing by 1. ✓
Watch out. Work out the rule from the terms you are given, and use it on the last term. Do not start again from the first term.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Find the rule that takes you from one term to the next, then use it to find the next 2 terms.
- Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.1.2 − 0.5 = 0.7
1.9 − 1.2 = 0.7
2.6 − 1.9 = 0.7 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 0.7, so the rule is add 710.
- Use the rule to carry on the sequence. Apply the rule to the last term you have, then to the new term, and so on.2.6 + 0.7 = 3.3
3.3 + 0.7 = 4 - Write down the new terms. 3.3, 4
Check. Jump straight there: 0.5 + 5 × 0.7 = 4, and the last term we found is 4. ✓
Watch out. Work out the rule from the terms you are given, and use it on the last term. Do not start again from the first term.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. To find a term-to-term rule, look at how each term changes into the next: first the differences, then the ratios.
- Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.11 − 6 = 5
16 − 11 = 5
21 − 16 = 5 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 5, so the rule is add 5.
Check. Start at 6 and apply the rule 3 times: 6 + 3 × 5 = 21, which is the last term. ✓
Watch out. The rule must work for every pair of neighbouring terms, not just the first pair. Always test it on the whole sequence.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. To find a term-to-term rule, look at how each term changes into the next: first the differences, then the ratios.
- Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.93 − 100 = −7
86 − 93 = −7
79 − 86 = −7 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is −7, so the rule is subtract 7.
Check. Start at 100 and apply the rule 3 times: 100 − 3 × 7 = 79, which is the last term. ✓
Watch out. The rule must work for every pair of neighbouring terms, not just the first pair. Always test it on the whole sequence.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. To find a term-to-term rule, look at how each term changes into the next: first the differences, then the ratios.
- Try the differences first. We always look for "add or subtract the same number" before anything else.6 − 3 = 3
12 − 6 = 6
24 − 12 = 12
The differences are not all the same, so it is not an add/subtract rule. - Now divide each term by the term before it. If we are multiplying by the same number each time, dividing a term by the one before gives that number.6 ÷ 3 = 2
12 ÷ 6 = 2
24 ÷ 12 = 2 - Look at the answers. They are all the same, so we multiply by that number each time.The rule is multiply by 2.
Check. Use the rule step by step from 3: 3, 6, 12, 24. Each term follows from the one before. ✓
Watch out. The rule must work for every pair of neighbouring terms, not just the first pair. Always test it on the whole sequence.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. To find a term-to-term rule, look at how each term changes into the next: first the differences, then the ratios.
- Try the differences first. We always look for "add or subtract the same number" before anything else.54 − 162 = −108
18 − 54 = −36
6 − 18 = −12
The differences are not all the same, so it is not an add/subtract rule. - Now divide each term by the term before it. If we are multiplying by the same number each time, dividing a term by the one before gives that number.54 ÷ 162 = 13
18 ÷ 54 = 13
6 ÷ 18 = 13 - Look at the answers. They are all the same, so we multiply by that number each time.The rule is divide by 3.
Check. Use the rule step by step from 162: 162, 54, 18, 6. Each term follows from the one before. ✓
Watch out. The rule must work for every pair of neighbouring terms, not just the first pair. Always test it on the whole sequence.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. To find a term-to-term rule, look at how each term changes into the next: first the differences, then the ratios.
- Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.3 − 2.5 = 0.5
3.5 − 3 = 0.5
4 − 3.5 = 0.5 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 0.5, so the rule is add 12.
Check. Start at 2.5 and apply the rule 3 times: 2.5 + 3 × 0.5 = 4, which is the last term. ✓
Watch out. The rule must work for every pair of neighbouring terms, not just the first pair. Always test it on the whole sequence.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. To find a term-to-term rule, look at how each term changes into the next: first the differences, then the ratios.
- Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.−1 − −4 = 3
2 − −1 = 3
5 − 2 = 3 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 3, so the rule is add 3.
Check. Start at −4 and apply the rule 3 times: −4 + 3 × 3 = 5, which is the last term. ✓
Watch out. The rule must work for every pair of neighbouring terms, not just the first pair. Always test it on the whole sequence.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. A term-to-term rule tells you what to do to one term to get the next. Start with the first term and apply the rule again and again.
- Write down the first term. The first term is given, so it is the start of the list.Term 1 = 4
- Apply the rule "add 6" to find each new term. Each term comes from the term before it. Use the answer you have just found as the next starting point.Term 2: 4 + 6 = 10
Term 3: 10 + 6 = 16
Term 4: 16 + 6 = 22
Term 5: 22 + 6 = 28 - Write the terms as a list. 4, 10, 16, 22, 28
Check. Jump straight to the last term: 4 + 4 × 6 = 28. ✓
Watch out. Do the operations in the order the rule gives them. "Multiply by 2 then add 1" is not the same as "add 1 then multiply by 2".
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. A term-to-term rule tells you what to do to one term to get the next. Start with the first term and apply the rule again and again.
- Write down the first term. The first term is given, so it is the start of the list.Term 1 = 100
- Apply the rule "subtract 7" to find each new term. Each term comes from the term before it. Use the answer you have just found as the next starting point.Term 2: 100 − 7 = 93
Term 3: 93 − 7 = 86
Term 4: 86 − 7 = 79
Term 5: 79 − 7 = 72 - Write the terms as a list. 100, 93, 86, 79, 72
Check. Jump straight to the last term: 100 − 4 × 7 = 72. ✓
Watch out. Do the operations in the order the rule gives them. "Multiply by 2 then add 1" is not the same as "add 1 then multiply by 2".
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. A term-to-term rule tells you what to do to one term to get the next. Start with the first term and apply the rule again and again.
- Write down the first term. The first term is given, so it is the start of the list.Term 1 = 3
- Apply the rule "multiply by 2" to find each new term. Each term comes from the term before it. Use the answer you have just found as the next starting point.Term 2: 3 × 2 = 6
Term 3: 6 × 2 = 12
Term 4: 12 × 2 = 24
Term 5: 24 × 2 = 48 - Write the terms as a list. 3, 6, 12, 24, 48
Check. Test the rule on the last pair: 24 × 2 gives 48. ✓
Watch out. Do the operations in the order the rule gives them. "Multiply by 2 then add 1" is not the same as "add 1 then multiply by 2".
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. A term-to-term rule tells you what to do to one term to get the next. Start with the first term and apply the rule again and again.
- Write down the first term. The first term is given, so it is the start of the list.Term 1 = 80
- Apply the rule "divide by 2" to find each new term. Each term comes from the term before it. Use the answer you have just found as the next starting point.Term 2: 80 ÷ 2 = 40
Term 3: 40 ÷ 2 = 20
Term 4: 20 ÷ 2 = 10
Term 5: 10 ÷ 2 = 5 - Write the terms as a list. 80, 40, 20, 10, 5
Check. Test the rule on the last pair: 10 ÷ 2 gives 5. ✓
Watch out. Do the operations in the order the rule gives them. "Multiply by 2 then add 1" is not the same as "add 1 then multiply by 2".
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. A term-to-term rule tells you what to do to one term to get the next. Start with the first term and apply the rule again and again.
- Write down the first term. The first term is given, so it is the start of the list.Term 1 = 2
- Apply the rule "multiply by 3 and then subtract 1" to find each new term. Each term comes from the term before it. Use the answer you have just found as the next starting point.Term 2: 2 × 3 = 6, then 6 − 1 = 5
Term 3: 5 × 3 = 15, then 15 − 1 = 14
Term 4: 14 × 3 = 42, then 42 − 1 = 41
Term 5: 41 × 3 = 123, then 123 − 1 = 122 - Write the terms as a list. 2, 5, 14, 41, 122
Check. Test the rule on the last pair: 41 × 3, then − 1 gives 122. ✓
Watch out. Do the operations in the order the rule gives them. "Multiply by 2 then add 1" is not the same as "add 1 then multiply by 2".
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. A term-to-term rule tells you what to do to one term to get the next. Start with the first term and apply the rule again and again.
- Write down the first term. The first term is given, so it is the start of the list.Term 1 = 1
- Apply the rule "multiply by 2 and then add 3" to find each new term. Each term comes from the term before it. Use the answer you have just found as the next starting point.Term 2: 1 × 2 = 2, then 2 + 3 = 5
Term 3: 5 × 2 = 10, then 10 + 3 = 13
Term 4: 13 × 2 = 26, then 26 + 3 = 29
Term 5: 29 × 2 = 58, then 58 + 3 = 61 - Write the terms as a list. 1, 5, 13, 29, 61
Check. Test the rule on the last pair: 29 × 2, then + 3 gives 61. ✓
Watch out. Do the operations in the order the rule gives them. "Multiply by 2 then add 1" is not the same as "add 1 then multiply by 2".
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. A term-to-term rule tells you what to do to one term to get the next. Start with the first term and apply the rule again and again.
- Write down the first term. The first term is given, so it is the start of the list.Term 1 = 8
- Apply the rule "add 5" to find each new term. Each term comes from the term before it. Use the answer you have just found as the next starting point.Term 2: 8 + 5 = 13
Term 3: 13 + 5 = 18
Term 4: 18 + 5 = 23
Term 5: 23 + 5 = 28
Term 6: 28 + 5 = 33 - Pick out term number 6. The 6th term is 33.
Check. Jump straight to the last term: 8 + 5 × 5 = 33. ✓
Watch out. Do the operations in the order the rule gives them. "Multiply by 2 then add 1" is not the same as "add 1 then multiply by 2".
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Find the rule that takes you from one term to the next, then use it to find the 10th term.
- Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.8 − 3 = 5
13 − 8 = 5
18 − 13 = 5 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 5, so the rule is add 5.
- Use the rule to carry on the sequence. Apply the rule to the last term you have, then to the new term, and so on.18 + 5 = 23
23 + 5 = 28
28 + 5 = 33
33 + 5 = 38
38 + 5 = 43
43 + 5 = 48 - Pick out the term you want. The 10th term is the one in position 10. Count along the list.3, 8, 13, 18, 23, 28, 33, 38, 43, 48
The 10th term is 48.
Check. Jump straight there: 3 + 9 × 5 = 48, and the last term we found is 48. ✓
Watch out. Work out the rule from the terms you are given, and use it on the last term. Do not start again from the first term.
Method. Test the rule on every pair of neighbouring terms.
- Check what “add 3” gives. If the rule is add 3 then starting from 3 the next term should be 6, and then 6 + 3 = 9.3 + 3 = 6 ✓ but 6 + 3 = 9, and the sequence says 12 ✗
- Look at the differences. Subtract each term from the next.6 − 3 = 3, 12 − 6 = 6, 24 − 12 = 12
The differences are 3, 6, 12. They are not the same, so it is not an “add the same number” rule. - Find the right rule. Try dividing each term by the one before it.6 ÷ 3 = 2, 12 ÷ 6 = 2, 24 ÷ 12 = 2
The rule is multiply by 2.
Check. 3 × 2 = 6, 6 × 2 = 12, 12 × 2 = 24. ✓
Watch out. A rule has to work for every pair of terms, not just the first pair.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Find the common difference using two terms you know, then use it to fill the gaps.
- Find the common difference from two terms you know. The terms in positions 1 and 3 are known. There are 2 steps between them, and each step adds the same amount.17 − 5 = 12
12 ÷ 2 = 6 for one step
Rule: add 6 - Fill in the gaps. Add the common difference to go forwards. Subtract it to go backwards.Term 2 = term 1 + 6 = 5 + 6 = 11
Term 5 = term 4 + 6 = 23 + 6 = 29 - Write the complete sequence. 5, 11, 17, 23, 29
Check. Look at the whole list: the difference between neighbours is 6 every time. ✓
Watch out. If a gap is at the start, work backwards by doing the opposite of the rule.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Find the common difference using two terms you know, then use it to fill the gaps.
- Find the common difference from two terms you know. The terms in positions 1 and 2 are known. There are 1 step between them, and each step adds the same amount.88 − 100 = −12
Rule: subtract 12 - Fill in the gaps. Add the common difference to go forwards. Subtract it to go backwards.Term 3 = term 2 − 12 = 88 − 12 = 76
Term 5 = term 4 − 12 = 64 − 12 = 52 - Write the complete sequence. 100, 88, 76, 64, 52
Check. Look at the whole list: the difference between neighbours is −12 every time. ✓
Watch out. If a gap is at the start, work backwards by doing the opposite of the rule.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Find the common difference using two terms you know, then use it to fill the gaps.
- Find the common difference from two terms you know. The terms in positions 2 and 3 are known. There are 1 step between them, and each step adds the same amount.21 − 14 = 7
Rule: add 7 - Fill in the gaps. Add the common difference to go forwards. Subtract it to go backwards.Term 1 = term 2 − 7 = 14 − 7 = 7
Term 4 = term 3 + 7 = 21 + 7 = 28 - Write the complete sequence. 7, 14, 21, 28, 35
Check. Look at the whole list: the difference between neighbours is 7 every time. ✓
Watch out. If a gap is at the start, work backwards by doing the opposite of the rule.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Find the common difference using two terms you know, then use it to fill the gaps.
- Find the common difference from two terms you know. The terms in positions 1 and 3 are known. There are 2 steps between them, and each step adds the same amount.−4 − (−12) = 8
8 ÷ 2 = 4 for one step
Rule: add 4 - Fill in the gaps. Add the common difference to go forwards. Subtract it to go backwards.Term 2 = term 1 + 4 = −12 + 4 = −8
Term 5 = term 4 + 4 = 0 + 4 = 4 - Write the complete sequence. −12, −8, −4, 0, 4
Check. Look at the whole list: the difference between neighbours is 4 every time. ✓
Watch out. If a gap is at the start, work backwards by doing the opposite of the rule.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Find the rule that takes you from one term to the next, then use it to find the next 2 terms.
- Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.12 − 14 = 14
34 − 12 = 14
1 − 34 = 14 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 14, so the rule is add 14.
- Use the rule to carry on the sequence. Apply the rule to the last term you have, then to the new term, and so on.1 + 14 = 54
54 + 14 = 32 - Write down the new terms. 54, 32
Check. Jump straight there: 14 + 5 × 14 = 32, and the last term we found is 32. ✓
Watch out. Work out the rule from the terms you are given, and use it on the last term. Do not start again from the first term.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Find the rule that takes you from one term to the next, then use it to find the next 2 terms.
- Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.1 − 12 = 12
32 − 1 = 12
2 − 32 = 12 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 12, so the rule is add 12.
- Use the rule to carry on the sequence. Apply the rule to the last term you have, then to the new term, and so on.2 + 12 = 52
52 + 12 = 3 - Write down the new terms. 52, 3
Check. Jump straight there: 12 + 5 × 12 = 3, and the last term we found is 3. ✓
Watch out. Work out the rule from the terms you are given, and use it on the last term. Do not start again from the first term.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Find the rule that takes you from one term to the next, then use it to find the next 2 terms.
- Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.5 − 8 = −3
2 − 5 = −3
−1 − 2 = −3 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is −3, so the rule is subtract 3.
- Use the rule to carry on the sequence. Apply the rule to the last term you have, then to the new term, and so on.−1 − 3 = −4
−4 − 3 = −7 - Write down the new terms. −4, −7
Check. Jump straight there: 8 − 5 × 3 = −7, and the last term we found is −7. ✓
Watch out. Work out the rule from the terms you are given, and use it on the last term. Do not start again from the first term.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Find the rule that takes you from one term to the next, then use it to find the next 2 terms.
- Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.−15 − −20 = 5
−10 − −15 = 5
−5 − −10 = 5 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 5, so the rule is add 5.
- Use the rule to carry on the sequence. Apply the rule to the last term you have, then to the new term, and so on.−5 + 5 = 0
0 + 5 = 5 - Write down the new terms. 0, 5
Check. Jump straight there: −20 + 5 × 5 = 5, and the last term we found is 5. ✓
Watch out. Work out the rule from the terms you are given, and use it on the last term. Do not start again from the first term.
Method. Look at how all the differences change, not just the last one.
- Work out the differences. Subtract each term from the next.2 − 1 = 1, 4 − 2 = 2, 7 − 4 = 3, 11 − 7 = 4
- Spot how the differences change. They are 1, 2, 3, 4: they go up by 1 each time.The next difference should be 5.
- Find the next term. Add the new difference to the last term.11 + 5 = 16, not 15
Check. The differences of the list 1, 2, 4, 7, 11, 16 are 1, 2, 3, 4, 5: a steady pattern. ✓
Watch out. Check the whole pattern of differences before deciding what the rule is.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. A term-to-term rule tells you what to do to one term to get the next. Start with the first term and apply the rule again and again.
- Write down the first term. The first term is given, so it is the start of the list.Term 1 = 640
- Apply the rule "divide by 2" to find each new term. Each term comes from the term before it. Use the answer you have just found as the next starting point.Term 2: 640 ÷ 2 = 320
Term 3: 320 ÷ 2 = 160
Term 4: 160 ÷ 2 = 80
Term 5: 80 ÷ 2 = 40 - Pick out term number 5. The 5th term is 40.
Check. Test the rule on the last pair: 80 ÷ 2 gives 40. ✓
Watch out. Do the operations in the order the rule gives them. "Multiply by 2 then add 1" is not the same as "add 1 then multiply by 2".
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. A term-to-term rule tells you what to do to one term to get the next. Start with the first term and apply the rule again and again.
- Write down the first term. The first term is given, so it is the start of the list.Term 1 = 5
- Apply the rule "multiply by 2 and then subtract 3" to find each new term. Each term comes from the term before it. Use the answer you have just found as the next starting point.Term 2: 5 × 2 = 10, then 10 − 3 = 7
Term 3: 7 × 2 = 14, then 14 − 3 = 11
Term 4: 11 × 2 = 22, then 22 − 3 = 19
Term 5: 19 × 2 = 38, then 38 − 3 = 35
Term 6: 35 × 2 = 70, then 70 − 3 = 67
Term 7: 67 × 2 = 134, then 134 − 3 = 131 - Pick out term number 7. The 7th term is 131.
Check. Test the rule on the last pair: 67 × 2, then − 3 gives 131. ✓
Watch out. Do the operations in the order the rule gives them. "Multiply by 2 then add 1" is not the same as "add 1 then multiply by 2".
Method. Keep applying the rule until the term goes over 1000.
- Write the first few terms. Use the rule on the last term each time.2, 2 × 2 − 1 = 3, 3 × 2 − 1 = 5, 5 × 2 − 1 = 9, 9 × 2 − 1 = 17, 17 × 2 − 1 = 33
- Carry on. Doubling and taking away 1 makes the terms grow quickly.33 × 2 − 1 = 65, then 129, then 257, then 513 (6th to 10th terms)
513 × 2 − 1 = 1025 - Decide. The 10th term is 513, which is less than 1000, and the 11th term is 1025, which is more.The first term bigger than 1000 is 1025, the 11th term.
Check. The terms are 3, 5, 9, 17, 33, … each is 1 more than a power of 2 (2 + 1 = 3, 4 + 1 = 5, 8 + 1 = 9, …); 1024 + 1 = 1025. ✓
Watch out. Check that the term before is not already over 1000.