GCSE Maths Sequences and nth Term (Higher): check-up and exam-style questions

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. 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 − 4 = 7
   18 − 11 = 7
   25 − 18 = 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.
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.
   25 + 7 = 32
   32 + 7 = 39
4. Write down the new terms.
   32, 39

Check. Jump straight there: 4 + 5 × 7 = 39, and the last term we found is 39. ✓

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.
   15 − 5 = 10
   45 − 15 = 30
   135 − 45 = 90
   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.
   15 ÷ 5 = 3
   45 ÷ 15 = 3
   135 ÷ 45 = 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 5: 5, 15, 45, 135. 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. 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 − 3
2. Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).
   1st term = 5 × 1 − 3 = 5 − 3 = 2
3. Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).
   2nd term = 5 × 2 − 3 = 10 − 3 = 7
4. Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).
   3rd term = 5 × 3 − 3 = 15 − 3 = 12
5. Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).
   4th term = 5 × 4 − 3 = 20 − 3 = 17
6. Write the terms as a list.
   2, 7, 12, 17

Check. The terms should go up by 5 each time (the number in front of n): 5, 5, 5. ✓

Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.

Key words. An arithmetic sequence goes up or down by the same amount every time. That amount is the common difference, written d. The nth term is a formula that gives any term from its position n.

Method. Use the common difference to start the nth term (it is the number in front of n), then work out what has to be added or subtracted.

1. Write the position numbers above the terms. The position is the place of each term in the list: 1st, 2nd, 3rd, and so on. The letter n stands for any position.

   Position n	1	2	3	4
   Term	7	11	15	19

2. Find the common difference d. Subtract each term from the next one. In an arithmetic sequence you always get the same answer, and that answer is called the common difference.
   11 − 7 = 4
   15 − 11 = 4
   19 − 15 = 4
   Common difference d = 4
3. Write the 4 times table underneath. Each time the position goes up by 1, the term changes by 4. That is just like the 4 times table: 4, 8, 12, … So the nth term must start with 4n.

   Position n	1	2	3	4
   Term	7	11	15	19
   4n	4	8	12	16

4. Compare the terms with the times table. The times table has the right steps but may start in the wrong place. Work out (term − times table) for each position. It should be the same number every time.

   Term	7	11	15	19
   4n	4	8	12	16
   Term − 4n	3	3	3	3

   The adjustment is always 3. So we must add 3 to 4n.
5. Put it together and write the nth term. The nth term is the times table part plus the adjustment.
   nth term = 4n + 3
   Shortcut for later: the adjustment 3 is the term before the first term, 7 − 4 = 3.

Check. Test the formula on terms we already know. n = 1: 4 × 1 + 3 = 4 + 3 = 7 ✓   n = 4: 4 × 4 + 3 = 16 + 3 = 19 ✓

Watch out. The number in front of n is the common difference, not the first term. Check with the first term: if your formula does not give it, something is wrong.

Key words. An arithmetic sequence goes up or down by the same amount every time. That amount is the common difference, written d. The nth term is a formula that gives any term from its position n.

Method. Use the common difference to start the nth term (it is the number in front of n), then work out what has to be added or subtracted.

1. Write the position numbers above the terms. The position is the place of each term in the list: 1st, 2nd, 3rd, and so on. The letter n stands for any position.

   Position n	1	2	3	4
   Term	30	27	24	21

2. Find the common difference d. Subtract each term from the next one. In an arithmetic sequence you always get the same answer, and that answer is called the common difference.
   27 − 30 = −3
   24 − 27 = −3
   21 − 24 = −3
   Common difference d = −3
3. Write the −3 times table underneath. Each time the position goes up by 1, the term changes by −3. That is just like the −3 times table: −3, −6, −9, … So the nth term must start with −3n.

   Position n	1	2	3	4
   Term	30	27	24	21
   −3n	−3	−6	−9	−12

4. Compare the terms with the times table. The times table has the right steps but may start in the wrong place. Work out (term − times table) for each position. It should be the same number every time.

   Term	30	27	24	21
   −3n	−3	−6	−9	−12
   Term − −3n	33	33	33	33

   The adjustment is always 33. So we must add 33 to −3n.
5. Put it together and write the nth term. The nth term is the times table part plus the adjustment.
   nth term = −3n + 33   (you can also write it as 33 − 3n)
   Shortcut for later: the adjustment 33 is the term before the first term, 30 − (−3) = 33.

Check. Test the formula on terms we already know. n = 1: −3 × 1 + 33 = −3 + 33 = 30 ✓   n = 4: −3 × 4 + 33 = −12 + 33 = 21 ✓

Watch out. This sequence is going down, so the common difference is negative and the number in front of n is negative. Check with 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. A number is in a sequence only if it is the term in a whole-number position. Set the nth term equal to the number and solve for n.

1. Understand what the question asks. A number is in the sequence only if some position n gives it. Positions are whole numbers: 1, 2, 3, … So we need to find out whether n comes out as a whole number.
   nth term = 3n + 1
   Is there a whole number n with 3n + 1 = 100?
2. Set the nth term equal to the number and solve for n. Subtract 1 on both sides to get the n term on its own, then divide by 3.
   3n + 1 = 100
   3n = 100 − 1 = 99
3. Divide by 3. The inverse of multiplying by 3 is dividing by 3.
   n = 99 ÷ 3 = 33
4. Decide. 33 is a whole number and it is at least 1, so it is a real position.
   Yes: 100 is the 33rd term.

Check. Substitute n = 33: 3 × 33 + 1 = 99 + 1 = 100, which is 100. ✓

Watch out. Do not just say "it goes up in 4s so it is in the sequence". A number can be close to a term and still not be one. Work out n and check that it is a whole number.

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. Use the nth-term rule for triangular numbers.

1. Remember what triangular numbers are. A triangular number is the number of dots in a triangle with rows of 1, 2, 3, … dots. The nth triangular number is 1 + 2 + … + n. Two copies of the same triangle fit together to make a rectangle n by (n + 1), so one copy is half of that: n(n + 1) ÷ 2.
   nth term = n(n + 1)2
2. Replace n by 12. We want the term in position 12, so use 12 in place of every n.
   12th term = 12 × (12 + 1) ÷ 2
3. Work it out. Follow the order of operations: brackets, powers, multiply and divide, then add and subtract.
   12th term = 12 × (12 + 1) ÷ 2 = 12 × 13 ÷ 2 = 156 ÷ 2 = 78

Check. Add the numbers 1 to 12 directly: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 = 78. ✓

Watch out. Multiply n by (n + 1) first, and only then divide 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. Every term is the sum of the previous two. To find a missing term, add the two before it, or subtract to work backwards.

1. Use the rule: each term is the sum of the two terms before it. Look for a gap where you know the two terms before it (add them) or the two terms after it (subtract).
   Terms: 2, 3, ?, 8, ?
2. Find term 3. Term 3 is the sum of the two terms before it.
   2 + 3 = 5
3. Find term 5. Term 5 is the sum of the two terms before it.
   5 + 8 = 13
4. Write the complete sequence.
   2, 3, 5, 8, 13

Check. Check every term is the sum of the two before it: 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13. ✓

Watch out. When working backwards, subtract: the missing term is the later term minus its neighbour.

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. Set up and solve an inequality to find how far along the sequence you must go, then pick the next whole number.

1. Write what we want as an inequality. We want the first term that is greater than 100. The terms must be greater than 100.
   7n − 4 > 100
2. Solve it like an equation. Add 4 on both sides, then divide both sides by 7.
   7n > 100 + 4 = 104
   n > 104 ÷ 7 = 1047
3. Choose the smallest whole number that works. Positions must be whole numbers. n has to be bigger than 1047, so take the next whole number up.
   n = 15
4. Work out the term and test the one before it. The term in position n must be greater than 100, and the term before it must not be.
   15th term = 7 × 15 − 4 = 105 − 4 = 101
   14th term = 7 × 14 − 4 = 98 − 4 = 94 (not greater than 100)

Check. 101 is greater than 100 and 94 is not. ✓

Watch out. Do not round n to the nearest whole number. You need the first whole number that makes the term big (or small) enough, which means rounding up.

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. Count the matchsticks, find the rule for the sequence of counts, and use the picture to see why it works.

1. Count the matchsticks in each pattern. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.

   Pattern number n	1	2	3
   Number of matchsticks	3	5	7

2. Find the common difference d. Subtract each term from the next one. In an arithmetic sequence you always get the same answer, and that answer is called the common difference.
   5 − 3 = 2
   7 − 5 = 2
   Common difference d = 2
3. Write the 2 times table underneath. Each time the position goes up by 1, the term changes by 2. That is just like the 2 times table: 2, 4, 6, … So the nth term must start with 2n.

   Position n	1	2	3
   Term	3	5	7
   2n	2	4	6

4. Compare the terms with the times table. The times table has the right steps but may start in the wrong place. Work out (term − times table) for each position. It should be the same number every time.

   Term	3	5	7
   2n	2	4	6
   Term − 2n	1	1	1

   The adjustment is always 1. So we must add 1 to 2n.
5. Put it together and write the nth term. The nth term is the times table part plus the adjustment.
   nth term = 2n + 1
   Shortcut for later: the adjustment 1 is the term before the first term, 3 − 2 = 1.
6. Check that the rule makes sense with the picture. A good formula can be explained from the drawing.
   The first triangle needs 3 matchsticks. Each new triangle shares a side with the last one, so it needs only 2 new matchsticks. After n triangles there are 2 matchsticks for each triangle, plus 1 extra for the first one: 2n + 1.

Check. Test the rule on patterns you can see. n = 1: 2 × 1 + 1 = 2 + 1 = 3 ✓   n = 3: 2 × 3 + 1 = 6 + 1 = 7 ✓

Watch out. Do not just keep adding on a drawing for a large pattern number. Find the nth term and substitute.

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. 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.
   13 − 8 = 5
   18 − 13 = 5
   23 − 18 = 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 5, so the rule is add 5.
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.
   23 + 5 = 28
   28 + 5 = 33
4. Write down the new terms.
   28, 33

Check. Jump straight there: 8 + 5 × 5 = 33, and the last term we found is 33. ✓

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.

Marks. 1 mark for finding the common difference 5; 1 mark for 28 and 33.

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 − 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.

Check. Use the rule step by step from 2: 2, 6, 18, 54. 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.

Marks. 1 mark for “multiply by 3”.

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 = 4n + 3
2. Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).
   1st term = 4 × 1 + 3 = 4 + 3 = 7
3. Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).
   2nd term = 4 × 2 + 3 = 8 + 3 = 11
4. Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).
   3rd term = 4 × 3 + 3 = 12 + 3 = 15
5. Write the terms as a list.
   7, 11, 15

Check. The terms should go up by 4 each time (the number in front of n): 4, 4. ✓

Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.

Marks. 1 mark for a correct substitution; 1 mark for 7, 11, 15.

Key words. An arithmetic sequence goes up or down by the same amount every time. That amount is the common difference, written d. The nth term is a formula that gives any term from its position n.

Method. Use the common difference to start the nth term (it is the number in front of n), then work out what has to be added or subtracted.

1. Write the position numbers above the terms. The position is the place of each term in the list: 1st, 2nd, 3rd, and so on. The letter n stands for any position.

   Position n	1	2	3	4
   Term	6	11	16	21

2. Find the common difference d. Subtract each term from the next one. In an arithmetic sequence you always get the same answer, and that answer is called the common difference.
   11 − 6 = 5
   16 − 11 = 5
   21 − 16 = 5
   Common difference d = 5
3. Write the 5 times table underneath. Each time the position goes up by 1, the term changes by 5. That is just like the 5 times table: 5, 10, 15, … So the nth term must start with 5n.

   Position n	1	2	3	4
   Term	6	11	16	21
   5n	5	10	15	20

4. Compare the terms with the times table. The times table has the right steps but may start in the wrong place. Work out (term − times table) for each position. It should be the same number every time.

   Term	6	11	16	21
   5n	5	10	15	20
   Term − 5n	1	1	1	1

   The adjustment is always 1. So we must add 1 to 5n.
5. Put it together and write the nth term. The nth term is the times table part plus the adjustment.
   nth term = 5n + 1
   Shortcut for later: the adjustment 1 is the term before the first term, 6 − 5 = 1.

Check. Test the formula on terms we already know. n = 1: 5 × 1 + 1 = 5 + 1 = 6 ✓   n = 4: 5 × 4 + 1 = 20 + 1 = 21 ✓

Watch out. The number in front of n is the common difference, not the first term. Check with the first term: if your formula does not give it, something is wrong.

Marks. 1 mark for 5n; 1 mark for 5n + 1.

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. A number is in a sequence only if it is the term in a whole-number position. Set the nth term equal to the number and solve for n.

1. Understand what the question asks. A number is in the sequence only if some position n gives it. Positions are whole numbers: 1, 2, 3, … So we need to find out whether n comes out as a whole number.
   nth term = 4n − 1
   Is there a whole number n with 4n − 1 = 150?
2. Set the nth term equal to the number and solve for n. Add 1 on both sides to get the n term on its own, then divide by 4.
   4n − 1 = 150
   4n = 150 + 1 = 151
3. Divide by 4. The inverse of multiplying by 4 is dividing by 4.
   n = 151 ÷ 4 = 37.75
4. Decide. 37.75 is not a whole number, so there is no such position.
   No: 150 is not in the sequence.

Check. The terms either side are the 37th term, 147, and the 38th term, 151. 150 falls in the gap between them, so it is skipped. ✓

Watch out. Do not just say "it goes up in 4s so it is in the sequence". A number can be close to a term and still not be one. Work out n and check that it is a whole number.

Marks. 1 mark for setting 4n − 1 = 150; 1 mark for n = 37.75; 1 mark for the conclusion (not a whole number).

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. Count the matchsticks, find the rule for the sequence of counts, and use the picture to see why it works.

1. Count the matchsticks in each pattern. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.

   Pattern number n	1	2	3
   Number of matchsticks	4	7	10

2. Find the common difference d. Subtract each term from the next one. In an arithmetic sequence you always get the same answer, and that answer is called the common difference.
   7 − 4 = 3
   10 − 7 = 3
   Common difference d = 3
3. Write the 3 times table underneath. Each time the position goes up by 1, the term changes by 3. That is just like the 3 times table: 3, 6, 9, … So the nth term must start with 3n.

   Position n	1	2	3
   Term	4	7	10
   3n	3	6	9

4. Compare the terms with the times table. The times table has the right steps but may start in the wrong place. Work out (term − times table) for each position. It should be the same number every time.

   Term	4	7	10
   3n	3	6	9
   Term − 3n	1	1	1

   The adjustment is always 1. So we must add 1 to 3n.
5. Put it together and write the nth term. The nth term is the times table part plus the adjustment.
   nth term = 3n + 1
   Shortcut for later: the adjustment 1 is the term before the first term, 4 − 3 = 1.
6. Check that the rule makes sense with the picture. A good formula can be explained from the drawing.
   The first square needs 4 matchsticks. Each new square is added on the end and shares one side with the square before it, so it needs only 3 new matchsticks. After n squares there are 3 matchsticks for each square, plus 1 extra for the very first side: 3n + 1.
7. Replace n by 40. Pattern 40 is the one in position 40.
   Pattern 40: 3 × 40 + 1 = 120 + 1 = 121

Check. Test the rule on patterns you can see. n = 1: 3 × 1 + 1 = 3 + 1 = 4 ✓   n = 3: 3 × 3 + 1 = 9 + 1 = 10 ✓

Watch out. Do not just keep adding on a drawing for a large pattern number. Find the nth term and substitute.

Marks. 1 mark for 3 added each time; 1 mark for 3n + 1; 1 mark for 121.

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. Every term is the sum of the previous two. To find a missing term, add the two before it, or subtract to work backwards.

1. Use the rule: each term is the sum of the two terms before it. Look for a gap where you know the two terms before it (add them) or the two terms after it (subtract).
   Terms: 3, 4, ?, 11, ?
2. Find term 3. Term 3 is the sum of the two terms before it.
   3 + 4 = 7
3. Find term 5. Term 5 is the sum of the two terms before it.
   7 + 11 = 18
4. Write the complete sequence.
   3, 4, 7, 11, 18

Check. Check every term is the sum of the two before it: 3 + 4 = 7, 4 + 7 = 11, 7 + 11 = 18. ✓

Watch out. When working backwards, subtract: the missing term is the later term minus its neighbour.

Marks. 1 mark for 7; 1 mark for 18.

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. Set up and solve an inequality to find how far along the sequence you must go, then pick the next whole number.

1. Write what we want as an inequality. We want the first term that is greater than 200. The terms must be greater than 200.
   7n − 4 > 200
2. Solve it like an equation. Add 4 on both sides, then divide both sides by 7.
   7n > 200 + 4 = 204
   n > 204 ÷ 7 = 2047
3. Choose the smallest whole number that works. Positions must be whole numbers. n has to be bigger than 2047, so take the next whole number up.
   n = 30
4. Work out the term and test the one before it. The term in position n must be greater than 200, and the term before it must not be.
   30th term = 7 × 30 − 4 = 210 − 4 = 206
   29th term = 7 × 29 − 4 = 203 − 4 = 199 (not greater than 200)

Check. 206 is greater than 200 and 199 is not. ✓

Watch out. Do not round n to the nearest whole number. You need the first whole number that makes the term big (or small) enough, which means rounding up.

Marks. 1 mark for 7n − 4 > 200; 1 mark for n > 29.1…; 1 mark for 206.

Key words. An arithmetic sequence goes up or down by the same amount every time. That amount is the common difference, written d. The nth term is a formula that gives any term from its position n.

Method. Use the common difference to start the nth term (it is the number in front of n), then work out what has to be added or subtracted.

1. Write the position numbers above the terms. The position is the place of each term in the list: 1st, 2nd, 3rd, and so on. The letter n stands for any position.

   Position n	1	2	3	4
   Term	42	54	66	78

2. Find the common difference d. Subtract each term from the next one. In an arithmetic sequence you always get the same answer, and that answer is called the common difference.
   54 − 42 = 12
   66 − 54 = 12
   78 − 66 = 12
   Common difference d = 12
3. Write the 12 times table underneath. Each time the position goes up by 1, the term changes by 12. That is just like the 12 times table: 12, 24, 36, … So the nth term must start with 12n.

   Position n	1	2	3	4
   Term	42	54	66	78
   12n	12	24	36	48

4. Compare the terms with the times table. The times table has the right steps but may start in the wrong place. Work out (term − times table) for each position. It should be the same number every time.

   Term	42	54	66	78
   12n	12	24	36	48
   Term − 12n	30	30	30	30

   The adjustment is always 30. So we must add 30 to 12n.
5. Put it together and write the nth term. The nth term is the times table part plus the adjustment.
   nth term = 12n + 30
   Shortcut for later: the adjustment 30 is the term before the first term, 42 − 12 = 30.
6. Replace n by 15 to find the 15th term. The 15th term is the one in position 15.
   15th term = 12 × 15 + 30 = 180 + 30 = 210

Check. Test the formula on terms we already know. n = 1: 12 × 1 + 30 = 12 + 30 = 42 ✓   n = 4: 12 × 4 + 30 = 48 + 30 = 78 ✓

Watch out. The number in front of n is the common difference, not the first term. Check with the first term: if your formula does not give it, something is wrong.

Marks. 1 mark for the common difference 12; 1 mark for 12n + 30; 1 mark for substituting 15; 1 mark for £210.

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.

1. Write down the first term. The first term is given, so it is the start of the list.
   Term 1 = 4
2. Apply the rule "multiply by 2 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: 4 × 2 = 8, then 8 − 1 = 7
   Term 3: 7 × 2 = 14, then 14 − 1 = 13
3. Write the terms as a list.
   4, 7, 13

Check. Test the rule on the last pair: 7 × 2, then − 1 gives 13. ✓

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".

Marks. 1 mark for 7; 1 mark for 13.

Key words. A quadratic sequence has an nth term with an n² in it, such as 2n² + 3. The first differences are the gaps between neighbouring terms. The second differences are the gaps between the first differences.

Method. Find the second differences. Half of them gives the number in front of n². Subtract that part from the terms, and the leftover is linear.

1. Write the terms and find the first differences. The first differences are the gaps between neighbouring terms (later minus earlier). If they were all the same it would be an ordinary arithmetic sequence.

   Position n	1	2	3	4	5
   Term	2	7	14	23	34

   First differences: 5, 7, 9, 11
   They are not all the same, so it is not a linear sequence.
2. Find the second differences. Now find the gaps between the first differences. If these are all the same, the sequence is quadratic (it has an n² term).
   Second differences: 2, 2, 2
   They are all 2, so the sequence is quadratic.
3. Halve the second difference to find a, the number in front of n². The squares 1, 4, 9, 16 have first differences 3, 5, 7 and second differences 2. So n² has second difference 2. A sequence an² has second difference 2a. So a is half of the second difference.
   a = 2 ÷ 2 = 1
   The nth term starts with n².
4. Subtract n² from each term. This removes the quadratic part. Work out n² for each position, then take it away from the term. What is left must be a linear (arithmetic) sequence.

   Position n	1	2	3	4	5
   n²	1	4	9	16	25
   n2	1	4	9	16	25
   Term	2	7	14	23	34
   Term − n2	1	3	5	7	9

5. Find the nth term of what is left. The leftover numbers form an arithmetic sequence. Find its common difference (that is the number in front of n), then the number to add, exactly as for a linear sequence.
   Leftover sequence: 1, 3, 5, 7, 9
   Common difference = 3 − 1 = 2, so it starts with 2n.
   Adjustment: 1 − 2 = −1
   Leftover nth term = 2n − 1
6. Put the two parts together. The nth term is the quadratic part plus the linear part.
   nth term = n² + (2n − 1) = n² + 2n − 1

Check. Test the formula. n = 2: 2² + 2 × 2 − 1 = 4 + 2 × 2 − 1 = 4 + 4 − 1 = 7 ✓   n = 5: 5² + 2 × 5 − 1 = 25 + 2 × 5 − 1 = 25 + 10 − 1 = 34 ✓

Watch out. a is half of the second difference, not the second difference itself. And remember to subtract an² (using each position) before finding the linear part.

Marks. 1 mark for second differences 2; 1 mark for n²; 1 mark for n² + 2n − 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 96 = 12 × r³
2. Divide to isolate the power of r. Divide both sides by 12.
   r³ = 96 ÷ 12 = 8
3. Take the cube root. We need the positive number that multiplies by itself 3 times to give 8. The inverse of raising to the power 3 is the cube root.
   r = 2³ = 8, so r = 2
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 = 12 ÷ 2 = 6
5. Write the nth term. Use first term × r^{n − 1}.
   nth term = 6 × 2^{n − 1}

Check. Test both given terms. n = 2: 6 × 2^{2 − 1} = 6 × 2¹ = 6 × 2 = 12 ✓   n = 5: 6 × 2^{5 − 1} = 6 × 2⁴ = 6 × 16 = 96 ✓

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

Marks. 1 mark for r³ = 8; 1 mark for r = 2; 1 mark for first term 6; 1 mark for 6 × 2^{n − 1}.

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√2 ÷ 3 = √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.
   3√2 ÷ 3 = √2
3. Multiply the last term by r to carry on. Use √k × √k = k whenever two matching surds multiply.
   3√2 × √2 = 3 × 1 × (√2 × √2) = 3 × 2 = 6
   
   6 × √2 = 6√2
4. Write down the new terms.
   6, 6√2

Check. Look at every second term: 3√2, 6√2. They should be 2 times apart: 3√2 × 2 = 6√2. ✓

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

Marks. 1 mark for 3√2; 1 mark for 6; 1 mark for 6√2.

Method. Write the two consecutive terms using n and n + 1, add them and factorise.

1. Write the two terms. The next term uses n + 1 in place of n.
   nth term = 2n + 1
   (n + 1)th term = 2(n + 1) + 1 = 2n + 3
2. Add them.
   (2n + 1) + (2n + 3) = 4n + 4
3. Factorise.
   4n + 4 = 4(n + 1), a multiple of 4

Check. n = 5: 11 + 13 = 24 = 4 × 6. ✓

Watch out. Use algebra so the proof works for every n.

Marks. 1 mark for 2n + 3; 1 mark for 4n + 4; 1 mark for 4(n + 1) and a conclusion.