GCSE Maths: Geometric sequences (Higher)

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.

1. 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.
2. 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
3. Look at the answers. They are all the same, so we multiply by that number each time.
   The rule is multiply by 3.
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.
   54 × 3 = 162
   162 × 3 = 486
5. 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.

1. 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.
2. 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
3. Look at the answers. They are all the same, so we multiply by that number each time.
   The rule is divide by 10.
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.
   1 ÷ 10 = 0.1
   0.1 ÷ 10 = 0.01
5. 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.

1. 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.
2. 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
3. Look at the answers. They are all the same, so we multiply by that number each time.
   The rule is multiply by 3.
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.
   108 × 3 = 324
   324 × 3 = 972
5. 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.

1. 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.
2. 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
3. Look at the answers. They are all the same, so we multiply by that number each time.
   The rule is divide by 2.
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.
   1.5 ÷ 2 = 0.75
   0.75 ÷ 2 = 0.375
5. 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.

1. 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.
2. 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
3. Look at the answers. They are all the same, so we multiply by that number each time.
   The rule is multiply by 1.5.
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.
   27 × 1.5 = 40.5
   40.5 × 1.5 = 60.75
5. 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.

1. 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.
2. 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
3. Look at the answers. They are all the same, so we multiply by that number each time.
   The rule is multiply by 5.
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.
   50 × 5 = 250
   250 × 5 = 1250
5. 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.

1. 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.
2. 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
3. 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.

1. 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.
2. 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
3. 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.

1. 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.
2. 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
3. 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.

1. 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.
2. 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
3. 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.

1. 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.
2. 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
3. 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.

1. 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.
2. 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
3. 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.

1. 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
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 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.

1. 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.
2. 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
3. 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.

1. 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.
2. 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
3. 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.

1. 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
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 −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.

1. 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.
2. 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
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.

1. 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
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 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.

1. 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
2. Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).
   1st term = 2¹ = 2
3. Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).
   2nd term = 2² = 4
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
5. Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).
   4th term = 2⁴ = 16
6. 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.

1. 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
2. Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).
   1st term = 5¹ = 5
3. Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).
   2nd term = 5² = 25
4. Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).
   3rd term = 5³ = 125
5. Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).
   4th term = 5⁴ = 625
6. 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.

1. 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
2. Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).
   1st term = 10¹ = 10
3. Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).
   2nd term = 10² = 100
4. Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).
   3rd term = 10³ = 1000
5. Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).
   4th term = 10⁴ = 10000
6. 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.

1. 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
2. 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
3. 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
4. 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
5. 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
6. 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.

1. Write the rule. This formula gives the term for any position n.
   nth term = 2ⁿ
2. The position we want is 8, so replace n by 8. Every n in the formula becomes the same number.
   8th term = 2⁸
3. Work it out, doing powers first, then multiplying, then adding or subtracting. This is the BIDMAS order.
   8th term = 2⁸ = 256
4. 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.

1. Write the rule. This formula gives the term for any position n.
   nth term = 3ⁿ
2. The position we want is 6, so replace n by 6. Every n in the formula becomes the same number.
   6th term = 3⁶
3. Work it out, doing powers first, then multiplying, then adding or subtracting. This is the BIDMAS order.
   6th term = 3⁶ = 729
4. 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.

1. Write the rule. This formula gives the term for any position n.
   nth term = 10ⁿ
2. The position we want is 5, so replace n by 5. Every n in the formula becomes the same number.
   5th term = 10⁵
3. Work it out, doing powers first, then multiplying, then adding or subtracting. This is the BIDMAS order.
   5th term = 10⁵ = 100000
4. 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.

1. Write the rule. This formula gives the term for any position n.
   nth term = (12)ⁿ
2. The position we want is 4, so replace n by 4. Every n in the formula becomes the same number.
   4th term = (12)⁴
3. Work it out, doing powers first, then multiplying, then adding or subtracting. This is the BIDMAS order.
   4th term = (12)⁴ = 116
4. 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 geometric sequence is made by multiplying by the same number every time. That number is the common ratio, written r. You find it by dividing any term by the term before it.

Method. For a geometric sequence the nth term is (first term) × (common ratio)^{n − 1}.

1. Find the common ratio r. Divide each term by the term before it. In a geometric sequence the answer is always the same.
   10 ÷ 5 = 2
   20 ÷ 10 = 2
   40 ÷ 20 = 2
   r = 2
2. Say how each term is built from the first term. The first term is 5. To get to term 2 we multiply by r once; to get to term 3 we multiply by r twice; and so on. The number of times we multiply is always one less than the position.
   Term 1 = 5
   Term 2 = 5 × 2 (multiply by r once)
   Term 3 = 5 × 2² (multiply by r 2 times) = 20
   Term 4 = 5 × 2³ (multiply by r 3 times) = 40
3. Write the nth term. To reach position n we multiply the first term by r exactly (n − 1) times.
   nth term = a × r^{n − 1} = 5 × 2^{n − 1}

Check. Test the formula. n = 1: 5 × 2^{1 − 1} = 5 × 2⁰ = 5 × 1 = 5 ✓   n = 4: 5 × 2^{4 − 1} = 5 × 2³ = 5 × 8 = 40 ✓

Watch out. The power is n − 1, not n, because the first term has not been multiplied at all. Check that your formula gives the first term when n = 1.

Key words. A geometric sequence is made by multiplying by the same number every time. That number is the common ratio, written r. You find it by dividing any term by the term before it.

Method. For a geometric sequence the nth term is (first term) × (common ratio)^{n − 1}.

1. Find the common ratio r. Divide each term by the term before it. In a geometric sequence the answer is always the same.
   6 ÷ 3 = 2
   12 ÷ 6 = 2
   24 ÷ 12 = 2
   r = 2
2. Say how each term is built from the first term. The first term is 3. To get to term 2 we multiply by r once; to get to term 3 we multiply by r twice; and so on. The number of times we multiply is always one less than the position.
   Term 1 = 3
   Term 2 = 3 × 2 (multiply by r once)
   Term 3 = 3 × 2² (multiply by r 2 times) = 12
   Term 4 = 3 × 2³ (multiply by r 3 times) = 24
3. Write the nth term. To reach position n we multiply the first term by r exactly (n − 1) times.
   nth term = a × r^{n − 1} = 3 × 2^{n − 1}

Check. Test the formula. n = 1: 3 × 2^{1 − 1} = 3 × 2⁰ = 3 × 1 = 3 ✓   n = 4: 3 × 2^{4 − 1} = 3 × 2³ = 3 × 8 = 24 ✓

Watch out. The power is n − 1, not n, because the first term has not been multiplied at all. Check that your formula gives the first term when n = 1.

Key words. A geometric sequence is made by multiplying by the same number every time. That number is the common ratio, written r. You find it by dividing any term by the term before it.

Method. For a geometric sequence the nth term is (first term) × (common ratio)^{n − 1}.

1. Find the common ratio r. Divide each term by the term before it. In a geometric sequence the answer is always the same.
   27 ÷ 81 = 13
   9 ÷ 27 = 13
   3 ÷ 9 = 13
   r = 13
2. Say how each term is built from the first term. The first term is 81. To get to term 2 we multiply by r once; to get to term 3 we multiply by r twice; and so on. The number of times we multiply is always one less than the position.
   Term 1 = 81
   Term 2 = 81 × (13) (multiply by r once)
   Term 3 = 81 × (13)² (multiply by r 2 times) = 9
   Term 4 = 81 × (13)³ (multiply by r 3 times) = 3
3. Write the nth term. To reach position n we multiply the first term by r exactly (n − 1) times.
   nth term = a × r^{n − 1} = 81 × (13)^{n − 1}

Check. Test the formula. n = 1: 81 × (13)^{1 − 1} = 81 × (13)⁰ = 81 × 1 = 81 ✓   n = 4: 81 × (13)^{4 − 1} = 81 × (13)³ = 81 × 127 = 3 ✓

Watch out. The power is n − 1, not n, because the first term has not been multiplied at all. Check that your formula gives the first term when n = 1.

Key words. A geometric sequence is made by multiplying by the same number every time. That number is the common ratio, written r. You find it by dividing any term by the term before it.

Method. For a geometric sequence the nth term is (first term) × (common ratio)^{n − 1}.

1. Find the common ratio r. Divide each term by the term before it. In a geometric sequence the answer is always the same.
   24 ÷ 16 = 32
   36 ÷ 24 = 32
   54 ÷ 36 = 32
   r = 32
2. Say how each term is built from the first term. The first term is 16. To get to term 2 we multiply by r once; to get to term 3 we multiply by r twice; and so on. The number of times we multiply is always one less than the position.
   Term 1 = 16
   Term 2 = 16 × (32) (multiply by r once)
   Term 3 = 16 × (32)² (multiply by r 2 times) = 36
   Term 4 = 16 × (32)³ (multiply by r 3 times) = 54
3. Write the nth term. To reach position n we multiply the first term by r exactly (n − 1) times.
   nth term = a × r^{n − 1} = 16 × (32)^{n − 1}

Check. Test the formula. n = 1: 16 × (32)^{1 − 1} = 16 × (32)⁰ = 16 × 1 = 16 ✓   n = 4: 16 × (32)^{4 − 1} = 16 × (32)³ = 16 × 278 = 54 ✓

Watch out. The power is n − 1, not n, because the first term has not been multiplied at all. Check that your formula gives the first term when n = 1.

Key words. A geometric sequence is made by multiplying by the same number every time. That number is the common ratio, written r. You find it by dividing any term by the term before it.

Method. The ratio between two terms is r raised to the number of steps between them. Divide, then take a root.

1. Count the steps from the 2nd term to the 5th term. Each step multiplies by the common ratio r.
   5 − 2 = 3 steps, so 162 = 6 × r³
2. Divide to isolate the power of r. Divide both sides by 6.
   r³ = 162 ÷ 6 = 27
3. Take the cube root. We need the positive number that multiplies by itself 3 times to give 27. The inverse of raising to the power 3 is the cube root.
   r = 3³ = 27, so r = 3
4. Go back to the first term. The 2nd term has been multiplied by r exactly 1 time from the first term, so divide the 2nd term by r that many times.
   first term = 6 ÷ 3 = 2
5. Write the nth term. Use first term × r^{n − 1}.
   nth term = 2 × 3^{n − 1}

Check. Test both given terms. n = 2: 2 × 3^{2 − 1} = 2 × 3¹ = 2 × 3 = 6 ✓   n = 5: 2 × 3^{5 − 1} = 2 × 3⁴ = 2 × 81 = 162 ✓

Watch out. The number of steps is the difference of the positions, not the difference of the terms.

Key words. A geometric sequence is made by multiplying by the same number every time. That number is the common ratio, written r. You find it by dividing any term by the term before it.

Method. The ratio between two terms is r raised to the number of steps between them. Divide, then take a root.

1. Count the steps from the 1st term to the 3rd term. Each step multiplies by the common ratio r.
   3 − 1 = 2 steps, so 36 = 4 × r²
2. Divide to isolate the power of r. Divide both sides by 4.
   r² = 36 ÷ 4 = 9
3. Take the square root. We need the positive number that multiplies by itself 2 times to give 9. The inverse of raising to the power 2 is the square root.
   r = 3² = 9, so r = 3
4. Go back to the first term. The given term is the first term.
   first term = 4
5. Write the nth term. Use first term × r^{n − 1}.
   nth term = 4 × 3^{n − 1}

Check. Test both given terms. n = 1: 4 × 3^{1 − 1} = 4 × 3⁰ = 4 × 1 = 4 ✓   n = 3: 4 × 3^{3 − 1} = 4 × 3² = 4 × 9 = 36 ✓

Watch out. The number of steps is the difference of the positions, not the difference of the terms.

Key words. A geometric sequence is made by multiplying by the same number every time. That number is the common ratio, written r. You find it by dividing any term by the term before it.

Method. The ratio between two terms is r raised to the number of steps between them. Divide, then take a root.

1. Count the steps from the 3rd term to the 4th term. Each step multiplies by the common ratio r.
   4 − 3 = 1 step, so 6 = 12 × r
2. Divide to isolate the power of r. Divide both sides by 12.
   r = 6 ÷ 12 = 12
   r = 12
3. Go back to the first term. The 3rd term has been multiplied by r exactly 2 times from the first term, so divide the 3rd term by r that many times.
   first term = 12 ÷ (12)² = 48
4. Write the nth term. Use first term × r^{n − 1}.
   nth term = 48 × (12)^{n − 1}

Check. Test both given terms. n = 3: 48 × (12)^{3 − 1} = 48 × (12)² = 48 × 14 = 12 ✓   n = 4: 48 × (12)^{4 − 1} = 48 × (12)³ = 48 × 18 = 6 ✓

Watch out. The number of steps is the difference of the positions, not the difference of the terms.

Key words. A geometric sequence is made by multiplying by the same number every time. That number is the common ratio, written r. You find it by dividing any term by the term before it.

Method. The ratio between two terms is r raised to the number of steps between them. Divide, then take a root.

1. Count the steps from the 2nd term to the 4th term. Each step multiplies by the common ratio r.
   4 − 2 = 2 steps, so 5 = 20 × r²
2. Divide to isolate the power of r. Divide both sides by 20.
   r² = 5 ÷ 20 = 14
3. Take the square root. We need the positive number that multiplies by itself 2 times to give 14. The inverse of raising to the power 2 is the square root.
   r = (12)² = 14, so r = 12
4. Go back to the first term. The 2nd term has been multiplied by r exactly 1 time from the first term, so divide the 2nd term by r that many times.
   first term = 20 ÷ (12) = 40
5. Write the nth term. Use first term × r^{n − 1}.
   nth term = 40 × (12)^{n − 1}

Check. Test both given terms. n = 2: 40 × (12)^{2 − 1} = 40 × (12)¹ = 40 × 12 = 20 ✓   n = 4: 40 × (12)^{4 − 1} = 40 × (12)³ = 40 × 18 = 5 ✓

Watch out. The number of steps is the difference of the positions, not the difference of the terms.

Key words. A surd is a root that does not work out exactly, such as √2 or √3. The key fact is √k × √k = k. For example √3 × √3 = 3, and 2√3 × √3 = 2 × 3 = 6.

Method. Find the common ratio by dividing, then multiply by it. When two matching surds multiply, the surd disappears: √k × √k = k.

1. Divide the second term by the first to find the common ratio. In a geometric sequence every term divided by the one before gives the same answer, the common ratio r.
   2√3 ÷ 2 = √3
   
   62√3: there is a surd on the bottom, so multiply the top and the bottom by √3 (this does not change the value).
   6 × √32√3 × √3 = 6√36 = √3
   r = √3
2. Check the ratio with the next pair. The ratio must be the same everywhere, so test it on the other terms.
   2√3 ÷ 2 = √3
   6 ÷ 2√3 = √3
   6√3 ÷ 6 = √3
3. Multiply the last term by r to carry on. Use √k × √k = k whenever two matching surds multiply.
   6√3 × √3 = 6 × 1 × (√3 × √3) = 6 × 3 = 18
   
   18 × √3 = 18√3
4. Write down the new terms.
   18, 18√3

Check. Look at every second term: 6√3, 18√3. They should be 3 times apart: 6√3 × 3 = 18√3. ✓

Watch out. Do not write √3 × √3 as √6 or as 2√3. It is exactly 3.

Key words. A surd is a root that does not work out exactly, such as √2 or √3. The key fact is √k × √k = k. For example √3 × √3 = 3, and 2√3 × √3 = 2 × 3 = 6.

Method. Find the common ratio by dividing, then multiply by it. When two matching surds multiply, the surd disappears: √k × √k = k.

1. Divide the second term by the first to find the common ratio. In a geometric sequence every term divided by the one before gives the same answer, the common ratio r.
   √2 ÷ 1 = √2
   
   2√2: there is a surd on the bottom, so multiply the top and the bottom by √2 (this does not change the value).
   2 × √2√2 × √2 = 2√22 = √2
   r = √2
2. Check the ratio with the next pair. The ratio must be the same everywhere, so test it on the other terms.
   √2 ÷ 1 = √2
   2 ÷ √2 = √2
   2√2 ÷ 2 = √2
3. Multiply the last term by r to carry on. Use √k × √k = k whenever two matching surds multiply.
   2√2 × √2 = 2 × 1 × (√2 × √2) = 2 × 2 = 4
   
   4 × √2 = 4√2
4. Write down the new terms.
   4, 4√2

Check. Look at every second term: 2√2, 4√2. They should be 2 times apart: 2√2 × 2 = 4√2. ✓

Watch out. Do not write √3 × √3 as √6 or as 2√3. It is exactly 3.

Key words. A surd is a root that does not work out exactly, such as √2 or √3. The key fact is √k × √k = k. For example √3 × √3 = 3, and 2√3 × √3 = 2 × 3 = 6.

Method. Find the common ratio by dividing, then multiply by it. When two matching surds multiply, the surd disappears: √k × √k = k.

1. Divide the second term by the first to find the common ratio. In a geometric sequence every term divided by the one before gives the same answer, the common ratio r.
   3√3: there is a surd on the bottom, so multiply the top and the bottom by √3 (this does not change the value).
   3 × √3√3 × √3 = 3√33 = √3
   
   3√3 ÷ 3 = √3
   r = √3
2. Check the ratio with the next pair. The ratio must be the same everywhere, so test it on the other terms.
   3 ÷ √3 = √3
   3√3 ÷ 3 = √3
3. Multiply the last term by r to carry on. Use √k × √k = k whenever two matching surds multiply.
   3√3 × √3 = 3 × 1 × (√3 × √3) = 3 × 3 = 9
   
   9 × √3 = 9√3
4. Write down the new terms.
   9, 9√3

Check. Look at every second term: 3√3, 9√3. They should be 3 times apart: 3√3 × 3 = 9√3. ✓

Watch out. Do not write √3 × √3 as √6 or as 2√3. It is exactly 3.

Key words. A surd is a root that does not work out exactly, such as √2 or √3. The key fact is √k × √k = k. For example √3 × √3 = 3, and 2√3 × √3 = 2 × 3 = 6.

Method. Find the common ratio by dividing, then multiply by it. When two matching surds multiply, the surd disappears: √k × √k = k.

1. Divide the second term by the first to find the common ratio. In a geometric sequence every term divided by the one before gives the same answer, the common ratio r.
   62√3: there is a surd on the bottom, so multiply the top and the bottom by √3 (this does not change the value).
   6 × √32√3 × √3 = 6√36 = √3
   
   6√3 ÷ 6 = √3
   r = √3
2. Check the ratio with the next pair. The ratio must be the same everywhere, so test it on the other terms.
   6 ÷ 2√3 = √3
   6√3 ÷ 6 = √3
   18 ÷ 6√3 = √3
3. Multiply the last term by r to carry on. Use √k × √k = k whenever two matching surds multiply.
   18 × √3 = 18√3
   
   18√3 × √3 = 18 × 1 × (√3 × √3) = 18 × 3 = 54
4. Write down the new terms.
   18√3, 54

Check. Look at every second term: 18, 54. They should be 3 times apart: 18 × 3 = 54. ✓

Watch out. Do not write √3 × √3 as √6 or as 2√3. It is exactly 3.

Key words. A surd is a root that does not work out exactly, such as √2 or √3. The key fact is √k × √k = k. For example √3 × √3 = 3, and 2√3 × √3 = 2 × 3 = 6.

Method. Find the common ratio by dividing, then multiply by it. When two matching surds multiply, the surd disappears: √k × √k = k.

1. Divide the second term by the first to find the common ratio. In a geometric sequence every term divided by the one before gives the same answer, the common ratio r.
   5√5 ÷ 5 = √5
   r = √5
2. Check the ratio with the next pair. The ratio must be the same everywhere, so test it on the other terms.
   5√5 ÷ 5 = √5
3. Multiply the last term by r to carry on. Use √k × √k = k whenever two matching surds multiply.
   5√5 × √5 = 5 × 1 × (√5 × √5) = 5 × 5 = 25
   
   25 × √5 = 25√5
4. Write down the new terms.
   25, 25√5

Check. Look at every second term: 5√5, 25√5. They should be 5 times apart: 5√5 × 5 = 25√5. ✓

Watch out. Do not write √3 × √3 as √6 or as 2√3. It is exactly 3.