GCSE Maths: Geometric sequences (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)
Geometric sequences
An arithmetic sequence is made by adding the same number each time. A geometric sequence is made by multiplying by the same number each time. That number is the common ratio, written r. For 3, 6, 12, 24, … every term is double the one before, so r = 2. For 81, 27, 9, 3, … each term is a third of the one before, so r = 13 (we can say “divide by 3”). The ratio can also be a decimal: 8, 12, 18, 27 has r = 1.5.
Finding the common ratio. Divide any term by the term before it: 12 ÷ 6 = 2. Check with another pair, because the ratio must be the same everywhere. Geometric sequences grow (or shrink) very fast, much faster than arithmetic ones. Doubling 2, 4, 8, 16, 32, 64, 128 reaches 128 in only seven terms.
Geometric sequences appear in real life: bacteria that double every hour, a ball that rebounds to half its previous height, a phone that loses 20% of its value each year (multiply by 0.8). Money growing with compound interest (Chapter 5) is also geometric.
Position-to-term rules. For the powers of a number the nth term is rn. For example 3n gives 3, 9, 27, 81, … and (12)n gives 12, 14, 18, …. Take care: 3n means 3 multiplied by itself n times, not 3 × n.
Key ideas
- A geometric sequence multiplies by the same common ratio r each time. Find r by dividing a term by the one before it.
- Divide by a number is the same as multiplying by a fraction: “divide by 2” means r = 12.
- The powers of a number have nth term rn: 2, 4, 8, 16, … is 2n.
- Higher: the nth term is a × rn − 1.
- Higher: √k × √k = k.
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 − 2 = 4
18 − 6 = 12
54 − 18 = 36
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 ÷ 2 = 3
18 ÷ 6 = 3
54 ÷ 18 = 3 - Look at the answers. They are all the same, so we multiply by that number each time.The rule is multiply by 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.54 × 3 = 162
162 × 3 = 486 - Write down the new terms. 162, 486
Check. Look back along the whole list: 18, 54, 162, 486. Every term is the one before multiplied by 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.
- Try the differences first. We always look for "add or subtract the same number" before anything else.40 − 80 = −40
20 − 40 = −20
10 − 20 = −10
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.40 ÷ 80 = 0.5
20 ÷ 40 = 0.5
10 ÷ 20 = 0.5 - 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.10 ÷ 2 = 5
5 ÷ 2 = 2.5 - Write down the new terms. 5, 2.5
Check. Look back along the whole list: 20, 10, 5, 2.5. 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 1 term.
- Try the differences first. We always look for "add or subtract the same number" before anything else.12 − 8 = 4
18 − 12 = 6
27 − 18 = 9
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.12 ÷ 8 = 1.5
18 ÷ 12 = 1.5
27 ÷ 18 = 1.5 - Look at the answers. They are all the same, so we multiply by that number each time.The rule is multiply by 1.5.
- 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.27 × 1.5 = 40.5
- Write down the new terms. 40.5
Check. Look back along the whole list: 12, 18, 27, 40.5. Every term is the one before multiplied by 1.5. ✓
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.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = 3n
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 31 = 3
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 32 = 9
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 33 = 27
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 34 = 81
- Replace n by 5 to find the 5th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).5th term = 35 = 243
- Write the terms as a list. 3, 9, 27, 81, 243
Check. Re-do the last one on a calculator: the 5th term should be 243. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = (12)n
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = (12)1 = 12
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = (12)2 = 14
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = (12)3 = 18
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = (12)4 = 116
- Write the terms as a list. 12, 14, 18, 116
Check. Re-do the last one on a calculator: the 4th term should be 116. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Watch out
Do not mix up the two kinds of sequence. 2, 4, 8, 16 is geometric (multiply by 2), but 2, 4, 6, 8 is arithmetic (add 2). Test the differences first; if they are not the same, test the ratios. And remember that “divide by 2” is a geometric rule with r = 12, even though the numbers go down.Exercise 11G Geometric sequences
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 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
8 − 4 = 4
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.2 ÷ 1 = 2
4 ÷ 2 = 2
8 ÷ 4 = 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.8 × 2 = 16
16 × 2 = 32 - Write down the new terms. 16, 32
Check. Look back along the whole list: 4, 8, 16, 32. 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.9 − 3 = 6
27 − 9 = 18
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.9 ÷ 3 = 3
27 ÷ 9 = 3 - Look at the answers. They are all the same, so we multiply by that number each time.The rule is multiply by 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.27 × 3 = 81
81 × 3 = 243 - Write down the new terms. 81, 243
Check. Look back along the whole list: 9, 27, 81, 243. Every term is the one before multiplied by 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.
- 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.128 − 256 = −128
64 − 128 = −64
32 − 64 = −32
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.128 ÷ 256 = 12
64 ÷ 128 = 12
32 ÷ 64 = 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.32 ÷ 2 = 16
16 ÷ 2 = 8 - Write down the new terms. 16, 8
Check. Look back along the whole list: 64, 32, 16, 8. 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.6 − 2 = 4
18 − 6 = 12
54 − 18 = 36
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 ÷ 2 = 3
18 ÷ 6 = 3
54 ÷ 18 = 3 - Look at the answers. They are all the same, so we multiply by that number each time.The rule is multiply by 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.54 × 3 = 162
162 × 3 = 486 - Write down the new terms. 162, 486
Check. Look back along the whole list: 18, 54, 162, 486. Every term is the one before multiplied by 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.
- Try the differences first. We always look for "add or subtract the same number" before anything else.100 − 1000 = −900
10 − 100 = −90
1 − 10 = −9
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.100 ÷ 1000 = 0.1
10 ÷ 100 = 0.1
1 ÷ 10 = 0.1 - Look at the answers. They are all the same, so we multiply by that number each time.The rule is divide by 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.1 ÷ 10 = 0.1
0.1 ÷ 10 = 0.01 - Write down the new terms. 0.1, 0.01
Check. Look back along the whole list: 10, 1, 0.1, 0.01. Every term is the one before divided by 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.12 − 4 = 8
36 − 12 = 24
108 − 36 = 72
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.12 ÷ 4 = 3
36 ÷ 12 = 3
108 ÷ 36 = 3 - Look at the answers. They are all the same, so we multiply by that number each time.The rule is multiply by 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.108 × 3 = 324
324 × 3 = 972 - Write down the new terms. 324, 972
Check. Look back along the whole list: 36, 108, 324, 972. Every term is the one before multiplied by 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.
- Try the differences first. We always look for "add or subtract the same number" before anything else.3 − 6 = −3
1.5 − 3 = −1.5
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.3 ÷ 6 = 0.5
1.5 ÷ 3 = 0.5 - 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.1.5 ÷ 2 = 0.75
0.75 ÷ 2 = 0.375 - Write down the new terms. 0.75, 0.375
Check. Look back along the whole list: 3, 1.5, 0.75, 0.375. 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.12 − 8 = 4
18 − 12 = 6
27 − 18 = 9
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.12 ÷ 8 = 1.5
18 ÷ 12 = 1.5
27 ÷ 18 = 1.5 - Look at the answers. They are all the same, so we multiply by that number each time.The rule is multiply by 1.5.
- 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.27 × 1.5 = 40.5
40.5 × 1.5 = 60.75 - Write down the new terms. 40.5, 60.75
Check. Look back along the whole list: 18, 27, 40.5, 60.75. Every term is the one before multiplied by 1.5. ✓
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.
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 − 2 = 8
50 − 10 = 40
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 ÷ 2 = 5
50 ÷ 10 = 5 - Look at the answers. They are all the same, so we multiply by that number each time.The rule is multiply by 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.50 × 5 = 250
250 × 5 = 1250 - Write down the new terms. 250, 1250
Check. Look back along the whole list: 10, 50, 250, 1250. Every term is the one before multiplied by 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.
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.10 − 2 = 8
50 − 10 = 40
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 ÷ 2 = 5
50 ÷ 10 = 5 - Look at the answers. They are all the same, so we multiply by that number each time.The rule is multiply by 5.
Check. Use the rule step by step from 2: 2, 10, 50. 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.
- Try the differences first. We always look for "add or subtract the same number" before anything else.6 − 4 = 2
9 − 6 = 3
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 ÷ 4 = 1.5
9 ÷ 6 = 1.5 - Look at the answers. They are all the same, so we multiply by that number each time.The rule is multiply by 1.5.
Check. Use the rule step by step from 4: 4, 6, 9. 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.4 − 1 = 3
16 − 4 = 12
64 − 16 = 48
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.4 ÷ 1 = 4
16 ÷ 4 = 4
64 ÷ 16 = 4 - Look at the answers. They are all the same, so we multiply by that number each time.The rule is multiply by 4.
Check. Use the rule step by step from 1: 1, 4, 16, 64. 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.16 − 64 = −48
4 − 16 = −12
1 − 4 = −3
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.16 ÷ 64 = 14
4 ÷ 16 = 14
1 ÷ 4 = 14 - Look at the answers. They are all the same, so we multiply by that number each time.The rule is divide by 4.
Check. Use the rule step by step from 64: 64, 16, 4, 1. 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.8 − 4 = 4
12 − 8 = 4
16 − 12 = 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.
Check. Start at 4 and apply the rule 3 times: 4 + 3 × 4 = 16, 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.8 − 4 = 4
16 − 8 = 8
32 − 16 = 16
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.8 ÷ 4 = 2
16 ÷ 8 = 2
32 ÷ 16 = 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 4: 4, 8, 16, 32. 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.30 − 90 = −60
10 − 30 = −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.30 ÷ 90 = 13
10 ÷ 30 = 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 90: 90, 30, 10. 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.60 − 90 = −30
30 − 60 = −30
0 − 30 = −30 - 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 −30, so the rule is subtract 30.
Check. Start at 90 and apply the rule 3 times: 90 − 3 × 30 = 0, 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.21 − 7 = 14
63 − 21 = 42
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.21 ÷ 7 = 3
63 ÷ 21 = 3 - Look at the answers. They are all the same, so we multiply by that number each time.The rule is multiply by 3.
Check. Use the rule step by step from 7: 7, 21, 63. 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.14 − 7 = 7
21 − 14 = 7
28 − 21 = 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 add 7.
Check. Start at 7 and apply the rule 3 times: 7 + 3 × 7 = 28, 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. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = 2n
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 21 = 2
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 22 = 4
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 23 = 8
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 24 = 16
- Write the terms as a list. 2, 4, 8, 16
Check. Re-do the last one on a calculator: the 4th term should be 16. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = 5n
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 51 = 5
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 52 = 25
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 53 = 125
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 54 = 625
- Write the terms as a list. 5, 25, 125, 625
Check. Re-do the last one on a calculator: the 4th term should be 625. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = 10n
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 101 = 10
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 102 = 100
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 103 = 1000
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 104 = 10000
- Write the terms as a list. 10, 100, 1000, 10000
Check. Re-do the last one on a calculator: the 4th term should be 10000. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = (12)n
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = (12)1 = 12
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = (12)2 = 14
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = (12)3 = 18
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = (12)4 = 116
- Write the terms as a list. 12, 14, 18, 116
Check. Re-do the last one on a calculator: the 4th term should be 116. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the nth-term rule by the position you want.
- Write the rule. This formula gives the term for any position n.nth term = 2n
- The position we want is 8, so replace n by 8. Every n in the formula becomes the same number.8th term = 28
- Work it out, doing powers first, then multiplying, then adding or subtracting. This is the BIDMAS order.8th term = 28 = 256
- Write the answer. The 8th term is 256.
Check. Check on a calculator: the 8th term is 256. ✓
Watch out. Substitute the position number for n. For 3n + 2 with n = 50 you work out 3 × 50 + 2 = 152, you do not multiply anything else by 50.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the nth-term rule by the position you want.
- Write the rule. This formula gives the term for any position n.nth term = 3n
- The position we want is 6, so replace n by 6. Every n in the formula becomes the same number.6th term = 36
- Work it out, doing powers first, then multiplying, then adding or subtracting. This is the BIDMAS order.6th term = 36 = 729
- Write the answer. The 6th term is 729.
Check. Check on a calculator: the 6th term is 729. ✓
Watch out. Substitute the position number for n. For 3n + 2 with n = 50 you work out 3 × 50 + 2 = 152, you do not multiply anything else by 50.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the nth-term rule by the position you want.
- Write the rule. This formula gives the term for any position n.nth term = 10n
- The position we want is 5, so replace n by 5. Every n in the formula becomes the same number.5th term = 105
- Work it out, doing powers first, then multiplying, then adding or subtracting. This is the BIDMAS order.5th term = 105 = 100000
- Write the answer. The 5th term is 100000.
Check. Check on a calculator: the 5th term is 100000. ✓
Watch out. Substitute the position number for n. For 3n + 2 with n = 50 you work out 3 × 50 + 2 = 152, you do not multiply anything else by 50.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the nth-term rule by the position you want.
- Write the rule. This formula gives the term for any position n.nth term = (12)n
- The position we want is 4, so replace n by 4. Every n in the formula becomes the same number.4th term = (12)4
- Work it out, doing powers first, then multiplying, then adding or subtracting. This is the BIDMAS order.4th term = (12)4 = 116
- Write the answer. The 4th term is 116.
Check. Check on a calculator: the 4th term is 116. ✓
Watch out. Substitute the position number for n. For 3n + 2 with n = 50 you work out 3 × 50 + 2 = 152, you do not multiply anything else by 50.
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 = 50
- 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: 50 × 2 = 100
Term 3: 100 × 2 = 200
Term 4: 200 × 2 = 400
Term 5: 400 × 2 = 800
Term 6: 800 × 2 = 1600
Term 7: 1600 × 2 = 3200 - Pick out term number 7. The 7th term is 3200.
Check. Test the rule on the last pair: 1600 × 2 gives 3200. ✓
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 = 160
- 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: 160 ÷ 2 = 80
Term 3: 80 ÷ 2 = 40
Term 4: 40 ÷ 2 = 20
Term 5: 20 ÷ 2 = 10 - Pick out term number 5. The 5th term is 10.
Check. Test the rule on the last pair: 20 ÷ 2 gives 10. ✓
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 5th term.
- Try the differences first. We always look for "add or subtract the same number" before anything else.400 − 500 = −100
320 − 400 = −80
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.400 ÷ 500 = 0.8
320 ÷ 400 = 0.8 - Look at the answers. They are all the same, so we multiply by that number each time.The rule is multiply by 0.8.
- 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.320 × 0.8 = 256
256 × 0.8 = 204.8 - Pick out the term you want. The 5th term is the one in position 5. Count along the list.500, 400, 320, 256, 204.8
The 5th term is 204.8.
Check. Look back along the whole list: 400, 320, 256, 204.8. Every term is the one before multiplied by 0.8. ✓
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 for a common difference and a common ratio.
- Work out the differences. Subtract each term from the next.2 − 1 = 1, 4 − 2 = 2, 7 − 4 = 3, 11 − 7 = 4
They are not the same, so it is not arithmetic. - Work out the ratios. Divide each term by the one before it.2 ÷ 1 = 2, 4 ÷ 2 = 2, 7 ÷ 4 = 1.75, 11 ÷ 7 = 1.57…
They are not the same, so it is not geometric. - Answer. A geometric sequence needs the same ratio every time.Liam is not right.
Check. A geometric sequence starting 1, 2 would continue 4, 8, 16, but the 4th term here is 7. ✓
Watch out. Going up quickly is not enough: the ratio has to be the same every time.
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 7th term.
- Try the differences first. We always look for "add or subtract the same number" before anything else.12 − 3 = 9
48 − 12 = 36
192 − 48 = 144
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.12 ÷ 3 = 4
48 ÷ 12 = 4
192 ÷ 48 = 4 - Look at the answers. They are all the same, so we multiply by that number each time.The rule is multiply by 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.192 × 4 = 768
768 × 4 = 3072
3072 × 4 = 12288 - Pick out the term you want. The 7th term is the one in position 7. Count along the list.3, 12, 48, 192, 768, 3072, 12288
The 7th term is 12288.
Check. Look back along the whole list: 192, 768, 3072, 12288. Every term is the one before multiplied by 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.
Method. Work out both 10th terms and compare them.
- Find the 10th term of 2n. Replace n by 10.2 × 10 = 20
- Find the 10th term of 2n. This is 2 multiplied by itself 10 times.210 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024
- Compare. 1024 is far bigger than 20, so the doubling sequence grows much faster than the add-2 sequence.
Check. Doubling: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 (10 terms). ✓
Watch out. 210 means ten 2s multiplied, not 2 × 10.