GCSE Maths: Using the nth term (Higher)

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 = 31?
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 = 31
   3n = 31 − 1 = 30
3. Divide by 3. The inverse of multiplying by 3 is dividing by 3.
   n = 30 ÷ 3 = 10
4. Decide. 10 is a whole number and it is at least 1, so it is a real position.
   Yes: 31 is the 10th term.

Check. Substitute n = 10: 3 × 10 + 1 = 30 + 1 = 31, which is 31. ✓

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. 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 = 50?
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 = 50
   3n = 50 − 1 = 49
3. Divide by 3. The inverse of multiplying by 3 is dividing by 3.
   n = 49 ÷ 3 = 493
4. Decide. 493 is not a whole number, so there is no such position.
   No: 50 is not in the sequence.

Check. The terms either side are the 16th term, 49, and the 17th term, 52. 50 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.

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. 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 = 151?
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 = 151
   3n = 151 − 1 = 150
3. Divide by 3. The inverse of multiplying by 3 is dividing by 3.
   n = 150 ÷ 3 = 50
4. Decide. 50 is a whole number and it is at least 1, so it is a real position.
   Yes: 151 is the 50th term.

Check. Substitute n = 50: 3 × 50 + 1 = 150 + 1 = 151, which is 151. ✓

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. 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. 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.
6. 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 + 3
   Is there a whole number n with 4n + 3 = 100?
7. Set the nth term equal to the number and solve for n. Subtract 3 on both sides to get the n term on its own, then divide by 4.
   4n + 3 = 100
   4n = 100 − 3 = 97
8. Divide by 4. The inverse of multiplying by 4 is dividing by 4.
   n = 97 ÷ 4 = 24.25
9. Decide. 24.25 is not a whole number, so there is no such position.
   No: 100 is not in the sequence.

Check. The terms either side are the 24th term, 99, and the 25th term, 103. 100 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.

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. 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	3	9	15	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.
   9 − 3 = 6
   15 − 9 = 6
   21 − 15 = 6
   Common difference d = 6
3. Write the 6 times table underneath. Each time the position goes up by 1, the term changes by 6. That is just like the 6 times table: 6, 12, 18, … So the nth term must start with 6n.

   Position n	1	2	3	4
   Term	3	9	15	21
   6n	6	12	18	24

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	9	15	21
   6n	6	12	18	24
   Term − 6n	−3	−3	−3	−3

   The adjustment is always −3. So we must subtract 3 from 6n.
5. Put it together and write the nth term. The nth term is the times table part plus the adjustment.
   nth term = 6n − 3
   Shortcut for later: the adjustment −3 is the term before the first term, 3 − 6 = −3.
6. 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 = 6n − 3
   Is there a whole number n with 6n − 3 = 99?
7. Set the nth term equal to the number and solve for n. Add 3 on both sides to get the n term on its own, then divide by 6.
   6n − 3 = 99
   6n = 99 + 3 = 102
8. Divide by 6. The inverse of multiplying by 6 is dividing by 6.
   n = 102 ÷ 6 = 17
9. Decide. 17 is a whole number and it is at least 1, so it is a real position.
   Yes: 99 is the 17th term.

Check. Substitute n = 17: 6 × 17 − 3 = 102 − 3 = 99, which is 99. ✓

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. 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. 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	2	5	8	11

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 − 2 = 3
   8 − 5 = 3
   11 − 8 = 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	2	5	8	11
   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	2	5	8	11
   3n	3	6	9	12
   Term − 3n	−1	−1	−1	−1

   The adjustment is always −1. So we must subtract 1 from 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, 2 − 3 = −1.
6. 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 = 61?
7. 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 3.
   3n − 1 = 61
   3n = 61 + 1 = 62
8. Divide by 3. The inverse of multiplying by 3 is dividing by 3.
   n = 62 ÷ 3 = 623
9. Decide. 623 is not a whole number, so there is no such position.
   No: 61 is not in the sequence.

Check. The terms either side are the 20th term, 59, and the 21st term, 62. 61 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.

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. 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	10	15	20	25

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.
   15 − 10 = 5
   20 − 15 = 5
   25 − 20 = 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	10	15	20	25
   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	10	15	20	25
   5n	5	10	15	20
   Term − 5n	5	5	5	5

   The adjustment is always 5. So we must add 5 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 + 5
   Shortcut for later: the adjustment 5 is the term before the first term, 10 − 5 = 5.
6. 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 = 5n + 5
   Is there a whole number n with 5n + 5 = 87?
7. Set the nth term equal to the number and solve for n. Subtract 5 on both sides to get the n term on its own, then divide by 5.
   5n + 5 = 87
   5n = 87 − 5 = 82
8. Divide by 5. The inverse of multiplying by 5 is dividing by 5.
   n = 82 ÷ 5 = 16.4
9. Decide. 16.4 is not a whole number, so there is no such position.
   No: 87 is not in the sequence.

Check. The terms either side are the 16th term, 85, and the 17th term, 90. 87 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.

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 = 20 − 3n
   Is there a whole number n with 20 − 3n = −10?
2. Set the nth term equal to the number and solve for n. Subtract 20 on both sides to get the n term on its own, then divide by −3.
   20 − 3n = −10
   −3n = −10 − 20 = −30
3. Divide by (−3). The inverse of multiplying by −3 is dividing by −3.
   n = −30 ÷ (−3) = 10
4. Decide. 10 is a whole number and it is at least 1, so it is a real position.
   Yes: −10 is the 10th term.

Check. Substitute n = 10: 20 − 3 × 10 = 20 − 30 = −10, which is −10. ✓

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. 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 = 20 − 3n
   Is there a whole number n with 20 − 3n = −11?
2. Set the nth term equal to the number and solve for n. Subtract 20 on both sides to get the n term on its own, then divide by −3.
   20 − 3n = −11
   −3n = −11 − 20 = −31
3. Divide by (−3). The inverse of multiplying by −3 is dividing by −3.
   n = −31 ÷ (−3) = 313
4. Decide. 313 is not a whole number, so there is no such position.
   No: −11 is not in the sequence.

Check. The terms either side are the 10th term, −10, and the 11th term, −13. −11 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.

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 = 20 − 3n
   Is there a whole number n with 20 − 3n = 2?
2. Set the nth term equal to the number and solve for n. Subtract 20 on both sides to get the n term on its own, then divide by −3.
   20 − 3n = 2
   −3n = 2 − 20 = −18
3. Divide by (−3). The inverse of multiplying by −3 is dividing by −3.
   n = −18 ÷ (−3) = 6
4. Decide. 6 is a whole number and it is at least 1, so it is a real position.
   Yes: 2 is the 6th term.

Check. Substitute n = 6: 20 − 3 × 6 = 20 − 18 = 2, which is 2. ✓

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. 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 = 20 − 3n
   Is there a whole number n with 20 − 3n = 0?
2. Set the nth term equal to the number and solve for n. Subtract 20 on both sides to get the n term on its own, then divide by −3.
   20 − 3n = 0
   −3n = 0 − 20 = −20
3. Divide by (−3). The inverse of multiplying by −3 is dividing by −3.
   n = −20 ÷ (−3) = 203
4. Decide. 203 is not a whole number, so there is no such position.
   No: 0 is not in the sequence.

Check. The terms either side are the 6th term, 2, and the 7th term, −1. 0 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.

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. 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 250. The terms must be greater than 250.
   4n + 1 > 250
2. Solve it like an equation. Subtract 1 on both sides, then divide both sides by 4.
   4n > 250 − 1 = 249
   n > 249 ÷ 4 = 62.25
3. Choose the smallest whole number that works. Positions must be whole numbers. n has to be bigger than 62.25, so take the next whole number up.
   n = 63
4. Work out the term and test the one before it. The term in position n must be greater than 250, and the term before it must not be.
   63rd term = 4 × 63 + 1 = 252 + 1 = 253
   62nd term = 4 × 62 + 1 = 248 + 1 = 249 (not greater than 250)

Check. 253 is greater than 250 and 249 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. 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 500. The terms must be greater than 500.
   9n + 2 > 500
2. Solve it like an equation. Subtract 2 on both sides, then divide both sides by 9.
   9n > 500 − 2 = 498
   n > 498 ÷ 9 = 1663
3. Choose the smallest whole number that works. Positions must be whole numbers. n has to be bigger than 1663, so take the next whole number up.
   n = 56
4. Work out the term and test the one before it. The term in position n must be greater than 500, and the term before it must not be.
   56th term = 9 × 56 + 2 = 504 + 2 = 506
   55th term = 9 × 55 + 2 = 495 + 2 = 497 (not greater than 500)

Check. 506 is greater than 500 and 497 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. 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 1000. The terms must be greater than 1000.
   2n + 5 > 1000
2. Solve it like an equation. Subtract 5 on both sides, then divide both sides by 2.
   2n > 1000 − 5 = 995
   n > 995 ÷ 2 = 497.5
3. Choose the smallest whole number that works. Positions must be whole numbers. n has to be bigger than 497.5, so take the next whole number up.
   n = 498
4. Work out the term and test the one before it. The term in position n must be greater than 1000, and the term before it must not be.
   498th term = 2 × 498 + 5 = 996 + 5 = 1001
   497th term = 2 × 497 + 5 = 994 + 5 = 999 (not greater than 1000)

Check. 1001 is greater than 1000 and 999 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. 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 less than 0. The terms must be less than 0.
   50 − 3n < 0
2. Solve it for n. The term is falling by 3 each time. Get the n term onto the side where it is positive, so we never have to divide by a negative.
   50 − 3n < 0
   50 < 3n   (add 3n to both sides)
   n > 50 ÷ 3 = 503
3. Choose the smallest whole number that works. Positions must be whole numbers. n has to be bigger than 503, so take the next whole number up.
   n = 17
4. Work out the term and test the one before it. The term in position n must be less than 0, and the term before it must not be.
   17th term = 50 − 3 × 17 = 50 − 51 = −1
   16th term = 50 − 3 × 16 = 50 − 48 = 2 (not less than 0)

Check. −1 is less than 0 and 2 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. 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 less than 0. The terms must be less than 0.
   42 − 5n < 0
2. Solve it for n. The term is falling by 5 each time. Get the n term onto the side where it is positive, so we never have to divide by a negative.
   42 − 5n < 0
   42 < 5n   (add 5n to both sides)
   n > 42 ÷ 5 = 8.4
3. Choose the smallest whole number that works. Positions must be whole numbers. n has to be bigger than 8.4, so take the next whole number up.
   n = 9
4. Work out the term and test the one before it. The term in position n must be less than 0, and the term before it must not be.
   9th term = 42 − 5 × 9 = 42 − 45 = −3
   8th term = 42 − 5 × 8 = 42 − 40 = 2 (not less than 0)

Check. −3 is less than 0 and 2 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. 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 less than 0. The terms must be less than 0.
   100 − 7n < 0
2. Solve it for n. The term is falling by 7 each time. Get the n term onto the side where it is positive, so we never have to divide by a negative.
   100 − 7n < 0
   100 < 7n   (add 7n to both sides)
   n > 100 ÷ 7 = 1007
3. Choose the smallest whole number that works. Positions must be whole numbers. n has to be bigger than 1007, 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 less than 0, and the term before it must not be.
   15th term = 100 − 7 × 15 = 100 − 105 = −5
   14th term = 100 − 7 × 14 = 100 − 98 = 2 (not less than 0)

Check. −5 is less than 0 and 2 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	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.

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.

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 50. Pattern 50 is the one in position 50.
   Pattern 50: 3 × 50 + 1 = 150 + 1 = 151

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.

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. 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.
7. Set the rule equal to the number and solve. We want the pattern that has 71 matchsticks. Undo the + 1, then divide by 2.
   2n + 1 = 71
   2n = 71 − 1 = 70
   n = 70 ÷ 2 = 35
8. Write the answer. The pattern number is the value of n.
   Pattern 35

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

1. Count the tiles 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 tiles	5	9	13

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.
   9 − 5 = 4
   13 − 9 = 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
   Term	5	9	13
   4n	4	8	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	5	9	13
   4n	4	8	12
   Term − 4n	1	1	1

   The adjustment is always 1. So we must add 1 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 + 1
   Shortcut for later: the adjustment 1 is the term before the first term, 5 − 4 = 1.
6. Check that the rule makes sense with the picture. A good formula can be explained from the drawing.
   There is always 1 tile in the middle. Then there is an arm going up, down, left and right, and each arm has n tiles. So the number of tiles is 4 arms of n tiles plus the 1 in the middle: 4n + 1.

Check. Test the rule on patterns you can see. n = 1: 4 × 1 + 1 = 4 + 1 = 5 ✓   n = 3: 4 × 3 + 1 = 12 + 1 = 13 ✓

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

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

1. Count the tiles 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 tiles	5	9	13

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.
   9 − 5 = 4
   13 − 9 = 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
   Term	5	9	13
   4n	4	8	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	5	9	13
   4n	4	8	12
   Term − 4n	1	1	1

   The adjustment is always 1. So we must add 1 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 + 1
   Shortcut for later: the adjustment 1 is the term before the first term, 5 − 4 = 1.
6. Check that the rule makes sense with the picture. A good formula can be explained from the drawing.
   There is always 1 tile in the middle. Then there is an arm going up, down, left and right, and each arm has n tiles. So the number of tiles is 4 arms of n tiles plus the 1 in the middle: 4n + 1.
7. Replace n by 20. Pattern 20 is the one in position 20.
   Pattern 20: 4 × 20 + 1 = 80 + 1 = 81

Check. Test the rule on patterns you can see. n = 1: 4 × 1 + 1 = 4 + 1 = 5 ✓   n = 3: 4 × 3 + 1 = 12 + 1 = 13 ✓

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

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

1. Count the tiles 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 tiles	5	9	13

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.
   9 − 5 = 4
   13 − 9 = 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
   Term	5	9	13
   4n	4	8	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	5	9	13
   4n	4	8	12
   Term − 4n	1	1	1

   The adjustment is always 1. So we must add 1 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 + 1
   Shortcut for later: the adjustment 1 is the term before the first term, 5 − 4 = 1.
6. Check that the rule makes sense with the picture. A good formula can be explained from the drawing.
   There is always 1 tile in the middle. Then there is an arm going up, down, left and right, and each arm has n tiles. So the number of tiles is 4 arms of n tiles plus the 1 in the middle: 4n + 1.
7. Set the rule equal to the number and solve. We want the pattern that has 101 tiles. Undo the + 1, then divide by 4.
   4n + 1 = 101
   4n = 101 − 1 = 100
   n = 100 ÷ 4 = 25
8. Write the answer. The pattern number is the value of n.
   Pattern 25

Check. Test the rule on patterns you can see. n = 1: 4 × 1 + 1 = 4 + 1 = 5 ✓   n = 3: 4 × 3 + 1 = 12 + 1 = 13 ✓

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

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

1. Count the orange tiles 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 orange tiles	8	12	16

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.
   12 − 8 = 4
   16 − 12 = 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
   Term	8	12	16
   4n	4	8	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	8	12	16
   4n	4	8	12
   Term − 4n	4	4	4

   The adjustment is always 4. So we must add 4 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 + 4
   Shortcut for later: the adjustment 4 is the term before the first term, 8 − 4 = 4.
6. Check that the rule makes sense with the picture. A good formula can be explained from the drawing.
   The blue square in the middle is n tiles by n tiles. The orange border has n tiles along each of the 4 sides, plus 1 tile in each of the 4 corners: 4n + 4.

Check. Test the rule on patterns you can see. n = 1: 4 × 1 + 4 = 4 + 4 = 8 ✓   n = 3: 4 × 3 + 4 = 12 + 4 = 16 ✓

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

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

1. Count the orange tiles 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 orange tiles	8	12	16

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.
   12 − 8 = 4
   16 − 12 = 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
   Term	8	12	16
   4n	4	8	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	8	12	16
   4n	4	8	12
   Term − 4n	4	4	4

   The adjustment is always 4. So we must add 4 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 + 4
   Shortcut for later: the adjustment 4 is the term before the first term, 8 − 4 = 4.
6. Check that the rule makes sense with the picture. A good formula can be explained from the drawing.
   The blue square in the middle is n tiles by n tiles. The orange border has n tiles along each of the 4 sides, plus 1 tile in each of the 4 corners: 4n + 4.
7. Replace n by 25. Pattern 25 is the one in position 25.
   Pattern 25: 4 × 25 + 4 = 100 + 4 = 104

Check. Test the rule on patterns you can see. n = 1: 4 × 1 + 4 = 4 + 4 = 8 ✓   n = 3: 4 × 3 + 4 = 12 + 4 = 16 ✓

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

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 = 2n + 3
   Is there a whole number n with 2n + 3 = 47?
2. Set the nth term equal to the number and solve for n. Subtract 3 on both sides to get the n term on its own, then divide by 2.
   2n + 3 = 47
   2n = 47 − 3 = 44
3. Divide by 2. The inverse of multiplying by 2 is dividing by 2.
   n = 44 ÷ 2 = 22
4. Decide. 22 is a whole number and it is at least 1, so it is a real position.
   Yes: 47 is the 22nd term.

Check. Substitute n = 22: 2 × 22 + 3 = 44 + 3 = 47, which is 47. ✓

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. 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 = 2n + 3
   Is there a whole number n with 2n + 3 = 50?
2. Set the nth term equal to the number and solve for n. Subtract 3 on both sides to get the n term on its own, then divide by 2.
   2n + 3 = 50
   2n = 50 − 3 = 47
3. Divide by 2. The inverse of multiplying by 2 is dividing by 2.
   n = 47 ÷ 2 = 23.5
4. Decide. 23.5 is not a whole number, so there is no such position.
   No: 50 is not in the sequence.

Check. The terms either side are the 23rd term, 49, and the 24th term, 51. 50 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.

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 + 2
   Is there a whole number n with 4n + 2 = 100?
2. Set the nth term equal to the number and solve for n. Subtract 2 on both sides to get the n term on its own, then divide by 4.
   4n + 2 = 100
   4n = 100 − 2 = 98
3. Divide by 4. The inverse of multiplying by 4 is dividing by 4.
   n = 98 ÷ 4 = 24.5
4. Decide. 24.5 is not a whole number, so there is no such position.
   No: 100 is not in the sequence.

Check. The terms either side are the 24th term, 98, and the 25th term, 102. 100 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.

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 + 2
   Is there a whole number n with 4n + 2 = 102?
2. Set the nth term equal to the number and solve for n. Subtract 2 on both sides to get the n term on its own, then divide by 4.
   4n + 2 = 102
   4n = 102 − 2 = 100
3. Divide by 4. The inverse of multiplying by 4 is dividing by 4.
   n = 100 ÷ 4 = 25
4. Decide. 25 is a whole number and it is at least 1, so it is a real position.
   Yes: 102 is the 25th term.

Check. Substitute n = 25: 4 × 25 + 2 = 100 + 2 = 102, which is 102. ✓

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. 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. Work out the common difference from the two terms (change in terms ÷ change in positions), then find what must be added.

1. Find how many steps there are between the two terms. Going from the 3rd term to the 7th term means taking steps of the common difference d. The number of steps is the difference between the positions.
   7 − 3 = 4 steps
2. Find the total change in the terms. This is how much the sequence has gone up (or down) over those steps.
   27 − 11 = 16
3. Divide to find the common difference. 4 equal steps make a total change of 16, so one step is the total divided by 4.
   d = 16 ÷ 4 = 4
4. Use the common difference to start the nth term. The nth term of an arithmetic sequence starts with dn.
   nth term = 4n + (something)
5. Put the 3rd term into the formula to find the something. We know that when n = 3 the term is 11. Let the missing number be c.
   4n + c with n = 3: 4 × 3 + c = 11
   12 + c = 11
   c = 11 − 12 = −1
6. Write the nth term.
   nth term = 4n − 1

Check. Test both terms. n = 3: 4 × 3 − 1 = 12 − 1 = 11 ✓   n = 7: 4 × 7 − 1 = 28 − 1 = 27 ✓

Watch out. The common difference is the change in the terms divided by the change in the positions, not divided by the change in the terms.

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. Work out the common difference from the two terms (change in terms ÷ change in positions), then find what must be added.

1. Find how many steps there are between the two terms. Going from the 2nd term to the 6th term means taking steps of the common difference d. The number of steps is the difference between the positions.
   6 − 2 = 4 steps
2. Find the total change in the terms. This is how much the sequence has gone up (or down) over those steps.
   21 − 5 = 16
3. Divide to find the common difference. 4 equal steps make a total change of 16, so one step is the total divided by 4.
   d = 16 ÷ 4 = 4
4. Use the common difference to start the nth term. The nth term of an arithmetic sequence starts with dn.
   nth term = 4n + (something)
5. Put the 2nd term into the formula to find the something. We know that when n = 2 the term is 5. Let the missing number be c.
   4n + c with n = 2: 4 × 2 + c = 5
   8 + c = 5
   c = 5 − 8 = −3
6. Write the nth term.
   nth term = 4n − 3

Check. Test both terms. n = 2: 4 × 2 − 3 = 8 − 3 = 5 ✓   n = 6: 4 × 6 − 3 = 24 − 3 = 21 ✓

Watch out. The common difference is the change in the terms divided by the change in the positions, not divided by the change in the terms.

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. Work out the common difference from the two terms (change in terms ÷ change in positions), then find what must be added.

1. Find how many steps there are between the two terms. Going from the 4th term to the 9th term means taking steps of the common difference d. The number of steps is the difference between the positions.
   9 − 4 = 5 steps
2. Find the total change in the terms. This is how much the sequence has gone up (or down) over those steps.
   5 − 20 = −15
3. Divide to find the common difference. 5 equal steps make a total change of −15, so one step is the total divided by 5.
   d = −15 ÷ 5 = −3
4. Use the common difference to start the nth term. The nth term of an arithmetic sequence starts with dn.
   nth term = −3n + (something)
5. Put the 4th term into the formula to find the something. We know that when n = 4 the term is 20. Let the missing number be c.
   −3n + c with n = 4: −3 × 4 + c = 20
   −12 + c = 20
   c = 20 − (−12) = 32
6. Write the nth term.
   nth term = −3n + 32

Check. Test both terms. n = 4: −3 × 4 + 32 = −12 + 32 = 20 ✓   n = 9: −3 × 9 + 32 = −27 + 32 = 5 ✓

Watch out. The common difference is the change in the terms divided by the change in the positions, not divided by the change in the terms.

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. Work out the common difference from the two terms (change in terms ÷ change in positions), then find what must be added.

1. Find how many steps there are between the two terms. Going from the 5th term to the 8th term means taking steps of the common difference d. The number of steps is the difference between the positions.
   8 − 5 = 3 steps
2. Find the total change in the terms. This is how much the sequence has gone up (or down) over those steps.
   21 − 12 = 9
3. Divide to find the common difference. 3 equal steps make a total change of 9, so one step is the total divided by 3.
   d = 9 ÷ 3 = 3
4. Use the common difference to start the nth term. The nth term of an arithmetic sequence starts with dn.
   nth term = 3n + (something)
5. Put the 5th term into the formula to find the something. We know that when n = 5 the term is 12. Let the missing number be c.
   3n + c with n = 5: 3 × 5 + c = 12
   15 + c = 12
   c = 12 − 15 = −3
6. Write the nth term.
   nth term = 3n − 3

Check. Test both terms. n = 5: 3 × 5 − 3 = 15 − 3 = 12 ✓   n = 8: 3 × 8 − 3 = 24 − 3 = 21 ✓

Watch out. The common difference is the change in the terms divided by the change in the positions, not divided by the change in the terms.

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. Make the two nth terms equal and solve for n.

1. Say what the question means. The two sequences have the same term in the same position n when their nth terms are equal. Write the one with the bigger number of ns first.
   8n − 12 = 5n + 3
2. Subtract 5n from both sides. This gets all the n terms onto one side. Doing the same to both sides keeps the equation balanced.
   3n − 12 = 3
3. Add 12 on both sides. This leaves the n term on its own.
   3n = 15
4. Divide both sides by 3.
   n = 15 ÷ 3 = 5
5. Check that n is a position. A position must be a positive whole number. It is.
   n = 5. Both sequences have the term 28 in position 5.

Check. Substitute: 5 × 5 + 3 = 25 + 3 = 28 and 8 × 5 − 12 = 40 − 12 = 28. ✓

Watch out. Subtract the smaller number of ns from both sides so that you keep a positive number of ns.

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. Make the two nth terms equal and solve for n.

1. Say what the question means. The two sequences have the same term in the same position n when their nth terms are equal. Write the one with the bigger number of ns first.
   6n − 5 = 4n + 1
2. Subtract 4n from both sides. This gets all the n terms onto one side. Doing the same to both sides keeps the equation balanced.
   2n − 5 = 1
3. Add 5 on both sides. This leaves the n term on its own.
   2n = 6
4. Divide both sides by 2.
   n = 6 ÷ 2 = 3
5. Check that n is a position. A position must be a positive whole number. It is.
   n = 3. Both sequences have the term 13 in position 3.

Check. Substitute: 4 × 3 + 1 = 12 + 1 = 13 and 6 × 3 − 5 = 18 − 5 = 13. ✓

Watch out. Subtract the smaller number of ns from both sides so that you keep a positive number of ns.

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. Make the two nth terms equal and solve for n.

1. Say what the question means. The two sequences have the same term in the same position n when their nth terms are equal. Write the one with the bigger number of ns first.
   5n + 2 = 3n + 7
2. Subtract 3n from both sides. This gets all the n terms onto one side. Doing the same to both sides keeps the equation balanced.
   2n + 2 = 7
3. Subtract 2 on both sides. This leaves the n term on its own.
   2n = 5
4. Divide both sides by 2.
   n = 5 ÷ 2 = 2.5
5. Check that n is a position. A position must be a positive whole number.
   2.5 is not a positive whole number, so the two sequences never have the same term in the same position.

Check. At position 2 the terms are 13 and 12; at position 3 they are 16 and 17. The sequences swap order without ever being equal. ✓

Watch out. Subtract the smaller number of ns from both sides so that you keep a positive number of ns.