GCSE Maths: Arithmetic sequences and the nth term (Higher)
Part of the GCSE Maths sequences and nth term chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Arithmetic sequences and the nth term
An arithmetic sequence (also called a linear sequence) goes up or down by the same amount every time. That amount is the common difference, written d. For 5, 8, 11, 14, … the common difference is 3. For 20, 17, 14, 11, … it is −3, because the sequence goes down. We would like a formula for the nth term, so that we can find any term without listing the others. Here is a method that works every time. We will find the nth term of 5, 8, 11, 14, …
Step 1. Find the common difference. 8 − 5 = 3, 11 − 8 = 3, 14 − 11 = 3. So d = 3.
Step 2. Write the times table of d and put it under the sequence. Each time n goes up by 1 the term goes up by 3, just like the 3 times table: 3, 6, 9, 12. So the nth term must start with 3n.
| Position n | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Sequence | 5 | 8 | 11 | 14 |
| 3n | 3 | 6 | 9 | 12 |
| Sequence − 3n | 2 | 2 | 2 | 2 |
Step 3. Compare. Subtract the times table from the sequence: 5 − 3 = 2, 8 − 6 = 2, 11 − 9 = 2, 14 − 12 = 2. The answer is always 2, so we must add 2.
Step 4. Write the nth term: 3n + 2. Check it: when n = 1, 3 × 1 + 2 = 5 ✓; when n = 4, 3 × 4 + 2 = 14 ✓.
Why does it work? The sequence starts at 5. If we went one step back from the first term we would reach 5 − 3 = 2. That “zeroth term” is exactly the number we add. So a quick way to find the adjustment is first term − d. In the picture the terms lie on a straight line, which is why these sequences are called linear. The common difference d is how steep the line is.
For a decreasing sequence the common difference is negative. For 20, 17, 14, 11, … we have d = −3, so the nth term starts with −3n. The times table is −3, −6, −9, −12. The sequence minus this is 23 each time, so the nth term is −3n + 23, which you may also write as 23 − 3n.
Some sequences you should know by heart: the even numbers 2, 4, 6, … have nth term 2n; the odd numbers 1, 3, 5, … have nth term 2n − 1; the multiples of 5 have nth term 5n. Once you have the nth term you can find any term, for example the 50th, by replacing n with 50.
Key ideas
- An arithmetic sequence has a common difference d (the same amount added each time).
- The nth term is dn + (adjustment), written like 3n + 2.
- Method: find d; write the d times table under the sequence; subtract to find the adjustment; write the nth term.
- The adjustment is also first term − d.
- Always check the formula with n = 1 and another position.
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.
- 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 5 8 11 14 - 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.8 − 5 = 3
11 − 8 = 3
14 − 11 = 3
Common difference d = 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 5 8 11 14 3n 3 6 9 12 - 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.
The adjustment is always 2. So we must add 2 to 3n.Term 5 8 11 14 3n 3 6 9 12 Term − 3n 2 2 2 2 - Put it together and write the nth term. The nth term is the times table part plus the adjustment.nth term = 3n + 2
Shortcut for later: the adjustment 2 is the term before the first term, 5 − 3 = 2.
Check. Test the formula on terms we already know. n = 1: 3 × 1 + 2 = 3 + 2 = 5 ✓ n = 4: 3 × 4 + 2 = 12 + 2 = 14 ✓
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.
- 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 5 Term 4 9 14 19 24 - 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 − 4 = 5
14 − 9 = 5
19 − 14 = 5
24 − 19 = 5
Common difference d = 5 - 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 5 Term 4 9 14 19 24 5n 5 10 15 20 25 - 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.
The adjustment is always −1. So we must subtract 1 from 5n.Term 4 9 14 19 24 5n 5 10 15 20 25 Term − 5n −1 −1 −1 −1 −1 - 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, 4 − 5 = −1.
Check. Test the formula on terms we already know. n = 1: 5 × 1 − 1 = 5 − 1 = 4 ✓ n = 5: 5 × 5 − 1 = 25 − 1 = 24 ✓
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.
- 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 20 17 14 11 - 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.17 − 20 = −3
14 − 17 = −3
11 − 14 = −3
Common difference d = −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 20 17 14 11 −3n −3 −6 −9 −12 - 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.
The adjustment is always 23. So we must add 23 to −3n.Term 20 17 14 11 −3n −3 −6 −9 −12 Term − −3n 23 23 23 23 - Put it together and write the nth term. The nth term is the times table part plus the adjustment.nth term = −3n + 23 (you can also write it as 23 − 3n)
Shortcut for later: the adjustment 23 is the term before the first term, 20 − (−3) = 23. - Replace n by 40 to find the 40th term. The 40th term is the one in position 40.40th term = −3 × 40 + 23 = −120 + 23 = −97
Check. Test the formula on terms we already know. n = 1: −3 × 1 + 23 = −3 + 23 = 20 ✓ n = 4: −3 × 4 + 23 = −12 + 23 = 11 ✓
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. 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. The nth term is (common difference) × n, plus an adjustment so that the first term comes out right.
- Write down what we know. The first term is the term when n = 1. The common difference is how much the sequence goes up (or down) each time.first term = 6, common difference d = 4
- Start the nth term with the common difference times n. Each step changes the term by 4, so the nth term starts with 4n.nth term = 4n + (adjustment)
- Find the adjustment. When n = 1 the term must be 6. The adjustment is whatever has to be added to 4n to make 6.4n with n = 1 gives 4
adjustment = 6 − 4 = 2 - Write the nth term. nth term = 4n + 2
Check. The first four terms should be 6, 10, 14, 18, … Test: n = 1: 4 × 1 + 2 = 4 + 2 = 6 ✓ n = 2: 4 × 2 + 2 = 8 + 2 = 10 ✓
Watch out. The number in front of n is the common difference, not the first term.
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.
- 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 12 17 22 - 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 − 7 = 5
17 − 12 = 5
22 − 17 = 5
Common difference d = 5 - 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 7 12 17 22 5n 5 10 15 20 - 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.
The adjustment is always 2. So we must add 2 to 5n.Term 7 12 17 22 5n 5 10 15 20 Term − 5n 2 2 2 2 - Put it together and write the nth term. The nth term is the times table part plus the adjustment.nth term = 5n + 2
Shortcut for later: the adjustment 2 is the term before the first term, 7 − 5 = 2. - Replace n by 50 to find the 50th term. The 50th term is the one in position 50.50th term = 5 × 50 + 2 = 250 + 2 = 252
Check. Test the formula on terms we already know. n = 1: 5 × 1 + 2 = 5 + 2 = 7 ✓ n = 4: 5 × 4 + 2 = 20 + 2 = 22 ✓
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.
- 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 32 2 52 3 - 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.2 − 32 = 12
52 − 2 = 12
3 − 52 = 12
Common difference d = 12 - Write the multiples of 12 underneath. Each time the position goes up by 1, the term changes by 12. That is just like the multiples of 12: 12, 1, 32, … So the nth term must start with 12n.
Position n 1 2 3 4 Term 32 2 52 3 12n 12 1 32 2 - 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.
The adjustment is always 1. So we must add 1 to 12n.Term 32 2 52 3 12n 12 1 32 2 Term − 12n 1 1 1 1 - Put it together and write the nth term. The nth term is the times table part plus the adjustment.nth term = 12n + 1
Shortcut for later: the adjustment 1 is the term before the first term, 32 − 12 = 1.
Check. Test the formula on terms we already know. n = 1: 12 × 1 + 1 = 12 + 1 = 32 ✓ n = 4: 12 × 4 + 1 = 2 + 1 = 3 ✓
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.
Watch out
The number in front of n is the common difference, not the first term. For 5, 8, 11, 14 the nth term is 3n + 2, not 5n + 3. Always test your formula with n = 1: it must give the first term.Exercise 11D The nth term of an arithmetic sequence
Non-calculatorKey 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.
- 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 4 7 10 13 - 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
13 − 10 = 3
Common difference d = 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 4 7 10 13 3n 3 6 9 12 - 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.
The adjustment is always 1. So we must add 1 to 3n.Term 4 7 10 13 3n 3 6 9 12 Term − 3n 1 1 1 1 - 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.
Check. Test the formula on terms we already know. n = 1: 3 × 1 + 1 = 3 + 1 = 4 ✓ n = 4: 3 × 4 + 1 = 12 + 1 = 13 ✓
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.
- 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 5 7 9 11 - 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 − 5 = 2
9 − 7 = 2
11 − 9 = 2
Common difference d = 2 - 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 4 Term 5 7 9 11 2n 2 4 6 8 - 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.
The adjustment is always 3. So we must add 3 to 2n.Term 5 7 9 11 2n 2 4 6 8 Term − 2n 3 3 3 3 - Put it together and write the nth term. The nth term is the times table part plus the adjustment.nth term = 2n + 3
Shortcut for later: the adjustment 3 is the term before the first term, 5 − 2 = 3.
Check. Test the formula on terms we already know. n = 1: 2 × 1 + 3 = 2 + 3 = 5 ✓ n = 4: 2 × 4 + 3 = 8 + 3 = 11 ✓
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.
- 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 8 13 18 - 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.8 − 3 = 5
13 − 8 = 5
18 − 13 = 5
Common difference d = 5 - 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 3 8 13 18 5n 5 10 15 20 - 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.
The adjustment is always −2. So we must subtract 2 from 5n.Term 3 8 13 18 5n 5 10 15 20 Term − 5n −2 −2 −2 −2 - Put it together and write the nth term. The nth term is the times table part plus the adjustment.nth term = 5n − 2
Shortcut for later: the adjustment −2 is the term before the first term, 3 − 5 = −2.
Check. Test the formula on terms we already know. n = 1: 5 × 1 − 2 = 5 − 2 = 3 ✓ n = 4: 5 × 4 − 2 = 20 − 2 = 18 ✓
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.
- 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 1 5 9 13 - 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 − 1 = 4
9 − 5 = 4
13 − 9 = 4
Common difference d = 4 - 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 1 5 9 13 4n 4 8 12 16 - 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.
The adjustment is always −3. So we must subtract 3 from 4n.Term 1 5 9 13 4n 4 8 12 16 Term − 4n −3 −3 −3 −3 - 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, 1 − 4 = −3.
Check. Test the formula on terms we already know. n = 1: 4 × 1 − 3 = 4 − 3 = 1 ✓ n = 4: 4 × 4 − 3 = 16 − 3 = 13 ✓
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.
- 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 20 30 40 - 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.20 − 10 = 10
30 − 20 = 10
40 − 30 = 10
Common difference d = 10 - Write the 10 times table underneath. Each time the position goes up by 1, the term changes by 10. That is just like the 10 times table: 10, 20, 30, … So the nth term must start with 10n.
Position n 1 2 3 4 Term 10 20 30 40 10n 10 20 30 40 - 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.
The adjustment is always 0. There is nothing to add or subtract: the sequence is the 10 times table.Term 10 20 30 40 10n 10 20 30 40 Term − 10n 0 0 0 0 - Put it together and write the nth term. The nth term is the times table part plus the adjustment.nth term = 10n
Shortcut for later: the adjustment 0 is the term before the first term, 10 − 10 = 0.
Check. Test the formula on terms we already know. n = 1: 10 × 1 = 10 ✓ n = 4: 10 × 4 = 40 ✓
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.
- 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 9 16 23 - 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 − 2 = 7
16 − 9 = 7
23 − 16 = 7
Common difference d = 7 - Write the 7 times table underneath. Each time the position goes up by 1, the term changes by 7. That is just like the 7 times table: 7, 14, 21, … So the nth term must start with 7n.
Position n 1 2 3 4 Term 2 9 16 23 7n 7 14 21 28 - 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.
The adjustment is always −5. So we must subtract 5 from 7n.Term 2 9 16 23 7n 7 14 21 28 Term − 7n −5 −5 −5 −5 - Put it together and write the nth term. The nth term is the times table part plus the adjustment.nth term = 7n − 5
Shortcut for later: the adjustment −5 is the term before the first term, 2 − 7 = −5.
Check. Test the formula on terms we already know. n = 1: 7 × 1 − 5 = 7 − 5 = 2 ✓ n = 4: 7 × 4 − 5 = 28 − 5 = 23 ✓
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.
- 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 20 17 14 11 - 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.17 − 20 = −3
14 − 17 = −3
11 − 14 = −3
Common difference d = −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 20 17 14 11 −3n −3 −6 −9 −12 - 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.
The adjustment is always 23. So we must add 23 to −3n.Term 20 17 14 11 −3n −3 −6 −9 −12 Term − −3n 23 23 23 23 - Put it together and write the nth term. The nth term is the times table part plus the adjustment.nth term = −3n + 23 (you can also write it as 23 − 3n)
Shortcut for later: the adjustment 23 is the term before the first term, 20 − (−3) = 23.
Check. Test the formula on terms we already know. n = 1: −3 × 1 + 23 = −3 + 23 = 20 ✓ n = 4: −3 × 4 + 23 = −12 + 23 = 11 ✓
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. 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.
- 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 26 22 18 - 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.26 − 30 = −4
22 − 26 = −4
18 − 22 = −4
Common difference d = −4 - 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 30 26 22 18 −4n −4 −8 −12 −16 - 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.
The adjustment is always 34. So we must add 34 to −4n.Term 30 26 22 18 −4n −4 −8 −12 −16 Term − −4n 34 34 34 34 - Put it together and write the nth term. The nth term is the times table part plus the adjustment.nth term = −4n + 34 (you can also write it as 34 − 4n)
Shortcut for later: the adjustment 34 is the term before the first term, 30 − (−4) = 34.
Check. Test the formula on terms we already know. n = 1: −4 × 1 + 34 = −4 + 34 = 30 ✓ n = 4: −4 × 4 + 34 = −16 + 34 = 18 ✓
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. 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.
- 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 50 45 40 35 - 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.45 − 50 = −5
40 − 45 = −5
35 − 40 = −5
Common difference d = −5 - 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 50 45 40 35 −5n −5 −10 −15 −20 - 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.
The adjustment is always 55. So we must add 55 to −5n.Term 50 45 40 35 −5n −5 −10 −15 −20 Term − −5n 55 55 55 55 - Put it together and write the nth term. The nth term is the times table part plus the adjustment.nth term = −5n + 55 (you can also write it as 55 − 5n)
Shortcut for later: the adjustment 55 is the term before the first term, 50 − (−5) = 55.
Check. Test the formula on terms we already know. n = 1: −5 × 1 + 55 = −5 + 55 = 50 ✓ n = 4: −5 × 4 + 55 = −20 + 55 = 35 ✓
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. 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.
- 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 9 7 5 3 - 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 − 9 = −2
5 − 7 = −2
3 − 5 = −2
Common difference d = −2 - 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 4 Term 9 7 5 3 −2n −2 −4 −6 −8 - 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.
The adjustment is always 11. So we must add 11 to −2n.Term 9 7 5 3 −2n −2 −4 −6 −8 Term − −2n 11 11 11 11 - Put it together and write the nth term. The nth term is the times table part plus the adjustment.nth term = −2n + 11 (you can also write it as 11 − 2n)
Shortcut for later: the adjustment 11 is the term before the first term, 9 − (−2) = 11.
Check. Test the formula on terms we already know. n = 1: −2 × 1 + 11 = −2 + 11 = 9 ✓ n = 4: −2 × 4 + 11 = −8 + 11 = 3 ✓
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. 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.
- 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 5 7 9 - 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
9 − 7 = 2
Common difference d = 2 - 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 4 Term 3 5 7 9 2n 2 4 6 8 - 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.
The adjustment is always 1. So we must add 1 to 2n.Term 3 5 7 9 2n 2 4 6 8 Term − 2n 1 1 1 1 - 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. - Replace n by 50 to find the 50th term. The 50th term is the one in position 50.50th term = 2 × 50 + 1 = 100 + 1 = 101
Check. Test the formula on terms we already know. n = 1: 2 × 1 + 1 = 2 + 1 = 3 ✓ n = 4: 2 × 4 + 1 = 8 + 1 = 9 ✓
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.
- 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 - 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 - 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 - 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.
The adjustment is always 1. So we must add 1 to 5n.Term 6 11 16 21 5n 5 10 15 20 Term − 5n 1 1 1 1 - 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. - Replace n by 30 to find the 30th term. The 30th term is the one in position 30.30th term = 5 × 30 + 1 = 150 + 1 = 151
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.
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.
- 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 8 14 20 - 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.8 − 2 = 6
14 − 8 = 6
20 − 14 = 6
Common difference d = 6 - 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 2 8 14 20 6n 6 12 18 24 - 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.
The adjustment is always −4. So we must subtract 4 from 6n.Term 2 8 14 20 6n 6 12 18 24 Term − 6n −4 −4 −4 −4 - Put it together and write the nth term. The nth term is the times table part plus the adjustment.nth term = 6n − 4
Shortcut for later: the adjustment −4 is the term before the first term, 2 − 6 = −4. - Replace n by 100 to find the 100th term. The 100th term is the one in position 100.100th term = 6 × 100 − 4 = 600 − 4 = 596
Check. Test the formula on terms we already know. n = 1: 6 × 1 − 4 = 6 − 4 = 2 ✓ n = 4: 6 × 4 − 4 = 24 − 4 = 20 ✓
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.
- 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 40 36 32 28 - 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.36 − 40 = −4
32 − 36 = −4
28 − 32 = −4
Common difference d = −4 - 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 40 36 32 28 −4n −4 −8 −12 −16 - 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.
The adjustment is always 44. So we must add 44 to −4n.Term 40 36 32 28 −4n −4 −8 −12 −16 Term − −4n 44 44 44 44 - Put it together and write the nth term. The nth term is the times table part plus the adjustment.nth term = −4n + 44 (you can also write it as 44 − 4n)
Shortcut for later: the adjustment 44 is the term before the first term, 40 − (−4) = 44. - Replace n by 12 to find the 12th term. The 12th term is the one in position 12.12th term = −4 × 12 + 44 = −48 + 44 = −4
Check. Test the formula on terms we already know. n = 1: −4 × 1 + 44 = −4 + 44 = 40 ✓ n = 4: −4 × 4 + 44 = −16 + 44 = 28 ✓
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. 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. The nth term is (common difference) × n, plus an adjustment so that the first term comes out right.
- Write down what we know. The first term is the term when n = 1. The common difference is how much the sequence goes up (or down) each time.first term = 7, common difference d = 3
- Start the nth term with the common difference times n. Each step changes the term by 3, so the nth term starts with 3n.nth term = 3n + (adjustment)
- Find the adjustment. When n = 1 the term must be 7. The adjustment is whatever has to be added to 3n to make 7.3n with n = 1 gives 3
adjustment = 7 − 3 = 4 - Write the nth term. nth term = 3n + 4
Check. The first four terms should be 7, 10, 13, 16, … Test: n = 1: 3 × 1 + 4 = 3 + 4 = 7 ✓ n = 2: 3 × 2 + 4 = 6 + 4 = 10 ✓
Watch out. The number in front of n is the common difference, not the first term.
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. The nth term is (common difference) × n, plus an adjustment so that the first term comes out right.
- Write down what we know. The first term is the term when n = 1. The common difference is how much the sequence goes up (or down) each time.first term = 2, common difference d = 5
- Start the nth term with the common difference times n. Each step changes the term by 5, so the nth term starts with 5n.nth term = 5n + (adjustment)
- Find the adjustment. When n = 1 the term must be 2. The adjustment is whatever has to be added to 5n to make 2.5n with n = 1 gives 5
adjustment = 2 − 5 = −3 - Write the nth term. nth term = 5n − 3
Check. The first four terms should be 2, 7, 12, 17, … Test: n = 1: 5 × 1 − 3 = 5 − 3 = 2 ✓ n = 2: 5 × 2 − 3 = 10 − 3 = 7 ✓
Watch out. The number in front of n is the common difference, not the first term.
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. The nth term is (common difference) × n, plus an adjustment so that the first term comes out right.
- Write down what we know. The first term is the term when n = 1. The common difference is how much the sequence goes up (or down) each time.first term = 20, common difference d = −4
- Start the nth term with the common difference times n. Each step changes the term by −4, so the nth term starts with −4n.nth term = −4n + (adjustment)
- Find the adjustment. When n = 1 the term must be 20. The adjustment is whatever has to be added to −4n to make 20.−4n with n = 1 gives −4
adjustment = 20 − (−4) = 24 - Write the nth term. nth term = −4n + 24 (or 24 − 4n)
Check. The first four terms should be 20, 16, 12, 8, … Test: n = 1: −4 × 1 + 24 = −4 + 24 = 20 ✓ n = 2: −4 × 2 + 24 = −8 + 24 = 16 ✓
Watch out. The number in front of n is the common difference, not the first term.
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. The nth term is (common difference) × n, plus an adjustment so that the first term comes out right.
- Write down what we know. The first term is the term when n = 1. The common difference is how much the sequence goes up (or down) each time.first term = 1, common difference d = 10
- Start the nth term with the common difference times n. Each step changes the term by 10, so the nth term starts with 10n.nth term = 10n + (adjustment)
- Find the adjustment. When n = 1 the term must be 1. The adjustment is whatever has to be added to 10n to make 1.10n with n = 1 gives 10
adjustment = 1 − 10 = −9 - Write the nth term. nth term = 10n − 9
Check. The first four terms should be 1, 11, 21, 31, … Test: n = 1: 10 × 1 − 9 = 10 − 9 = 1 ✓ n = 2: 10 × 2 − 9 = 20 − 9 = 11 ✓
Watch out. The number in front of n is the common difference, not the first term.
Method. Test Ben's rule with n = 1, then find the correct rule.
- Test Ben's rule with n = 1. The 1st term should be 6.6 × 1 + 4 = 10, but the 1st term is 6 ✗
- Find the common difference. This is the number in front of n.10 − 6 = 4, 14 − 10 = 4: d = 4
- Find the adjustment. First term minus d.6 − 4 = 2
- Write the nth term. 4n + 2
Check. 4 × 1 + 2 = 6 ✓ and 4 × 4 + 2 = 18 ✓
Watch out. Ben used the first term as the number in front of n. It should be the common difference.
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.
- 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 12 17 22 27 - 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.17 − 12 = 5
22 − 17 = 5
27 − 22 = 5
Common difference d = 5 - 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 12 17 22 27 5n 5 10 15 20 - 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.
The adjustment is always 7. So we must add 7 to 5n.Term 12 17 22 27 5n 5 10 15 20 Term − 5n 7 7 7 7 - Put it together and write the nth term. The nth term is the times table part plus the adjustment.nth term = 5n + 7
Shortcut for later: the adjustment 7 is the term before the first term, 12 − 5 = 7. - Replace n by 20 to find the 20th term. The 20th term is the one in position 20.20th term = 5 × 20 + 7 = 100 + 7 = 107
Check. Test the formula on terms we already know. n = 1: 5 × 1 + 7 = 5 + 7 = 12 ✓ n = 4: 5 × 4 + 7 = 20 + 7 = 27 ✓
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.
- 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 10 14 18 - 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.10 − 6 = 4
14 − 10 = 4
18 − 14 = 4
Common difference d = 4 - 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 6 10 14 18 4n 4 8 12 16 - 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.
The adjustment is always 2. So we must add 2 to 4n.Term 6 10 14 18 4n 4 8 12 16 Term − 4n 2 2 2 2 - Put it together and write the nth term. The nth term is the times table part plus the adjustment.nth term = 4n + 2
Shortcut for later: the adjustment 2 is the term before the first term, 6 − 4 = 2. - Replace n by 25 to find the 25th term. The 25th term is the one in position 25.25th term = 4 × 25 + 2 = 100 + 2 = 102
Check. Test the formula on terms we already know. n = 1: 4 × 1 + 2 = 4 + 2 = 6 ✓ n = 4: 4 × 4 + 2 = 16 + 2 = 18 ✓
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.
- 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 9 13 17 21 - 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.13 − 9 = 4
17 − 13 = 4
21 − 17 = 4
Common difference d = 4 - 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 9 13 17 21 4n 4 8 12 16 - 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.
The adjustment is always 5. So we must add 5 to 4n.Term 9 13 17 21 4n 4 8 12 16 Term − 4n 5 5 5 5 - Put it together and write the nth term. The nth term is the times table part plus the adjustment.nth term = 4n + 5
Shortcut for later: the adjustment 5 is the term before the first term, 9 − 4 = 5. - Replace n by 10 to find the 10th term. The 10th term is the one in position 10.10th term = 4 × 10 + 5 = 40 + 5 = 45
Check. Test the formula on terms we already know. n = 1: 4 × 1 + 5 = 4 + 5 = 9 ✓ n = 4: 4 × 4 + 5 = 16 + 5 = 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.
Method. Check that 4n + 3 produces the sequence.
- Substitute the first four positions. Work out 4n + 3 for n = 1, 2, 3, 4.n = 1: 4 × 1 + 3 = 7
n = 2: 4 × 2 + 3 = 11
n = 3: 4 × 3 + 3 = 15
n = 4: 4 × 4 + 3 = 19 - Explain why it carries on. When n goes up by 1, 4n goes up by 4, so the terms go up by 4 each time, just like the sequence.The sequence goes up by 4 each time and starts at 7, so 4n + 3 is the nth term.
Check. The common difference is 4 and 7 − 4 = 3, the number added. ✓
Watch out. In a “show that” question, write a conclusion sentence at the end.
Method. Substitute a few values into both rules and compare.
- Try n = 1 and n = 2 in Ali's rule. 3 × 1 + 5 = 8, 3 × 2 + 5 = 11
- Try the same values in Beth's rule. 5 + 3 × 1 = 8, 5 + 3 × 2 = 11
- Explain. Addition can be done in either order, so 3n + 5 = 5 + 3n.Both rules give the sequence 8, 11, 14, … so both are right.
Check. Both rules give 8, 11, 14, 17 for n = 1, 2, 3, 4. ✓
Watch out. The order of adding does not matter, but the order of subtracting does: 5 − 3n is not 3n − 5.
Method. The first term is when n = 1. The common difference is the number in front of n.
- Find the first term. Replace n by 1.4 × 1 − 1 = 3
- Find the second term to see the difference. Replace n by 2.4 × 2 − 1 = 7, and 7 − 3 = 4
- Read off the common difference. It is the number multiplying n.Common difference = 4
Check. The terms 3, 7, 11, … go up by 4. ✓
Watch out. The first term is not the same as the number in front of n.
Method. Split the nth term into a multiple of 3 plus 2, then test 100.
- Look at 3n. 3 times any whole number is a multiple of 3.3n is a multiple of 3
- Add the 2. Every term is a multiple of 3 plus 2, so dividing a term by 3 leaves remainder 2.Terms: 5, 8, 11, 14, … (5 = 3 × 1 + 2, 8 = 3 × 2 + 2)
- Test 100. Divide 100 by 3.100 ÷ 3 = 33 remainder 1 (since 3 × 33 = 99)
- Conclude. The remainder is 1, not 2.100 is not a term of the sequence.
Check. The terms near 100 are 3 × 32 + 2 = 98 and 3 × 33 + 2 = 101, which skip 100. ✓
Watch out. Looking at remainders is a quick way to rule numbers out.
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.
- 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 32 2 52 3 - 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.2 − 32 = 12
52 − 2 = 12
3 − 52 = 12
Common difference d = 12 - Write the multiples of 12 underneath. Each time the position goes up by 1, the term changes by 12. That is just like the multiples of 12: 12, 1, 32, … So the nth term must start with 12n.
Position n 1 2 3 4 Term 32 2 52 3 12n 12 1 32 2 - 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.
The adjustment is always 1. So we must add 1 to 12n.Term 32 2 52 3 12n 12 1 32 2 Term − 12n 1 1 1 1 - Put it together and write the nth term. The nth term is the times table part plus the adjustment.nth term = 12n + 1
Shortcut for later: the adjustment 1 is the term before the first term, 32 − 12 = 1.
Check. Test the formula on terms we already know. n = 1: 12 × 1 + 1 = 12 + 1 = 32 ✓ n = 4: 12 × 4 + 1 = 2 + 1 = 3 ✓
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.
- 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 72 4 92 5 - 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.4 − 72 = 12
92 − 4 = 12
5 − 92 = 12
Common difference d = 12 - Write the multiples of 12 underneath. Each time the position goes up by 1, the term changes by 12. That is just like the multiples of 12: 12, 1, 32, … So the nth term must start with 12n.
Position n 1 2 3 4 Term 72 4 92 5 12n 12 1 32 2 - 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.
The adjustment is always 3. So we must add 3 to 12n.Term 72 4 92 5 12n 12 1 32 2 Term − 12n 3 3 3 3 - Put it together and write the nth term. The nth term is the times table part plus the adjustment.nth term = 12n + 3
Shortcut for later: the adjustment 3 is the term before the first term, 72 − 12 = 3.
Check. Test the formula on terms we already know. n = 1: 12 × 1 + 3 = 12 + 3 = 72 ✓ n = 4: 12 × 4 + 3 = 2 + 3 = 5 ✓
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.
- 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 72 4 92 - 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.72 − 3 = 12
4 − 72 = 12
92 − 4 = 12
Common difference d = 12 - Write the multiples of 12 underneath. Each time the position goes up by 1, the term changes by 12. That is just like the multiples of 12: 12, 1, 32, … So the nth term must start with 12n.
Position n 1 2 3 4 Term 3 72 4 92 12n 12 1 32 2 - 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.
The adjustment is always 52. So we must add 52 to 12n.Term 3 72 4 92 12n 12 1 32 2 Term − 12n 52 52 52 52 - Put it together and write the nth term. The nth term is the times table part plus the adjustment.nth term = 12n + 52
Shortcut for later: the adjustment 52 is the term before the first term, 3 − 12 = 52.
Check. Test the formula on terms we already know. n = 1: 12 × 1 + 52 = 12 + 52 = 3 ✓ n = 4: 12 × 4 + 52 = 2 + 52 = 92 ✓
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.
- 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 13 23 1 43 - 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.23 − 13 = 13
1 − 23 = 13
43 − 1 = 13
Common difference d = 13 - Write the multiples of 13 underneath. Each time the position goes up by 1, the term changes by 13. That is just like the multiples of 13: 13, 23, 1, … So the nth term must start with 13n.
Position n 1 2 3 4 Term 13 23 1 43 13n 13 23 1 43 - 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.
The adjustment is always 0. There is nothing to add or subtract: the sequence is the multiples of 13.Term 13 23 1 43 13n 13 23 1 43 Term − 13n 0 0 0 0 - Put it together and write the nth term. The nth term is the times table part plus the adjustment.nth term = 13n
Shortcut for later: the adjustment 0 is the term before the first term, 13 − 13 = 0.
Check. Test the formula on terms we already know. n = 1: 13 × 1 = 13 ✓ n = 4: 13 × 4 = 43 ✓
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.
Method. In an arithmetic sequence the differences are equal. Make an equation from the two differences.
- Write the two differences. Second term minus first term, and third term minus second term.(2x + 1) − x = x + 1
(4x − 3) − (2x + 1) = 2x − 4 - Set them equal and solve. The two differences must be the same.x + 1 = 2x − 4
1 + 4 = 2x − x
x = 5 - Work out the terms. Replace x by 5.5, 11, 17
- Find the nth term. The common difference is 6, and 5 − 6 = −1.nth term = 6n − 1
Check. The terms 5, 11, 17 go up by 6 each time, and 6 × 1 − 1 = 5. ✓
Watch out. Do not forget to put x = 5 back in to find the actual terms.