Year 6: The nth term of a linear sequence
Part of the Year 6 algebra chapter. Chapter overview · whole chapter on one page
The nth term of a linear sequence
The nth term is a rule that gives any term of a sequence. The letter n is the position of the term. The first term has n = 1, the second term has n = 2, and so on.
For a linear sequence, the number in front of n is the difference. For 5, 8, 11, 14 the difference is 3, so the rule starts with 3n. Then we find the number to add so that the first term is right: 3 × 1 = 3, but the first term is 5, so we add 2. The nth term is 3n + 2.
We can also use the nth term to test whether a number is in the sequence. Solve the equation, and check that n is a whole number.
| Position n | Term | Working |
|---|---|---|
| 1 | 5 | 3 × 1 + 2 = 5 |
| 2 | 8 | 3 × 2 + 2 = 8 |
| 3 | 11 | 3 × 3 + 2 = 11 |
| 4 | 14 | 3 × 4 + 2 = 14 |
Key ideas
- The number in front of n is the difference between terms.
- Put n = 1 into the rule to check that the first term is right.
- To find a term in position n, put that number in place of n.
- To test a number, solve the equation. n must be a whole number that is 1 or more.
Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. Find the difference: it is the number in front of n. Then find the number that makes the first term right.
- Find the difference between two terms next to each other. It is the same every time, because the sequence is linear.8 - 5 = 3
- Start the rule with 3n, because the difference is 3. The number in front of n is the difference.3n + ?
- Find the number that is added: put n = 1 in the rule. The first term is 5, so the rule must give 5 when n = 1.3 × 1 + ? = 5, so ? = 5 - 3 = 2
- Check with the second term. Put n = 2 in the rule: the answer must be 8.3 × 2 + 2 = 8. ✓
- Write the nth term. 3n + 2
Check. Put n = 2 in 3n + 2: this gives the second term of the sequence. ✓
Watch out. The number in front of n is the difference, but the rule also needs the number that is added (or taken away).
Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.
Method. Put the number in place of the letter, multiply, then add or take away.
- Write the formula. The letter stands for a number we know, so we put it in.3n + 2
- Put 30 in place of n. Multiply first. Multiplication comes before addition and subtraction.3 × 30 = 90
- Add 2. This is the last step of the formula.90 + 2 = 92
- Write the answer. 92
Check. Work it out again: 3 × 30 = 90, then 90 + 2 = 92. ✓
Watch out. Multiply before you add or take away. Do not do the sum from left to right.
Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. Solve the equation for n. If n is a whole number that is 1 or more, the number is a term.
- Write the equation. We ask: for which n is the term equal to 101?4n + 3 = 101
- Take 3 away from both sides. This leaves the part with n.4n = 98
- Divide both sides by 4. This gives n, the position of the term.n = 98 ÷ 4 = 49/2
- Decide. The position must be a whole number, 1 or more.n = 49/2, which is not a whole number, so 101 is not a term
- Write the answer. No
Check. Try the nearest whole numbers: 4 × 24 + 3 and 4 × 25 + 3 are not 101. ✓
Watch out. The position n must be a whole number. A number between two terms is never a term.
Watch out
In 3n + 2 the first term is not 3. Put n = 1 in: 3 × 1 + 2 = 5. Always check the rule with n = 1 before you use it.
Exam tip
A descending sequence can be written with a minus sign in front of n. For 35, 30, 25 the nth term is 40 - 5n, because the difference is 5.
Exercise 8D Writing and using the nth term
Non-calculatorKey words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. Find the difference: it is the number in front of n. Then find the number that makes the first term right.
- Find the difference between two terms next to each other. It is the same every time, because the sequence is linear.5 - 2 = 3
- Start the rule with 3n, because the difference is 3. The number in front of n is the difference.3n + ?
- Find the number that is added: put n = 1 in the rule. The first term is 2, so the rule must give 2 when n = 1.3 × 1 + ? = 2, so ? = 2 - 3 = -1
- Check with the second term. Put n = 2 in the rule: the answer must be 5.3 × 2 - 1 = 5. ✓
- Write the nth term. 3n - 1
Check. Put n = 2 in 3n - 1: this gives the second term of the sequence. ✓
Watch out. The number in front of n is the difference, but the rule also needs the number that is added (or taken away).
Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. Find the difference: it is the number in front of n. Then find the number that makes the first term right.
- Find the difference between two terms next to each other. It is the same every time, because the sequence is linear.11 - 7 = 4
- Start the rule with 4n, because the difference is 4. The number in front of n is the difference.4n + ?
- Find the number that is added: put n = 1 in the rule. The first term is 7, so the rule must give 7 when n = 1.4 × 1 + ? = 7, so ? = 7 - 4 = 3
- Check with the second term. Put n = 2 in the rule: the answer must be 11.4 × 2 + 3 = 11. ✓
- Write the nth term. 4n + 3
Check. Put n = 2 in 4n + 3: this gives the second term of the sequence. ✓
Watch out. The number in front of n is the difference, but the rule also needs the number that is added (or taken away).
Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. Find the difference: it is the number in front of n. Then find the number that makes the first term right.
- Find the difference between two terms next to each other. It is the same every time, because the sequence is linear.13 - 10 = 3
- Start the rule with 3n, because the difference is 3. The number in front of n is the difference.3n + ?
- Find the number that is added: put n = 1 in the rule. The first term is 10, so the rule must give 10 when n = 1.3 × 1 + ? = 10, so ? = 10 - 3 = 7
- Check with the second term. Put n = 2 in the rule: the answer must be 13.3 × 2 + 7 = 13. ✓
- Write the nth term. 3n + 7
Check. Put n = 2 in 3n + 7: this gives the second term of the sequence. ✓
Watch out. The number in front of n is the difference, but the rule also needs the number that is added (or taken away).
Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. Find the difference: it is the number in front of n. Then find the number that makes the first term right.
- Find the difference between two terms next to each other. It is the same every time, because the sequence is linear.5 - 1 = 4
- Start the rule with 4n, because the difference is 4. The number in front of n is the difference.4n + ?
- Find the number that is added: put n = 1 in the rule. The first term is 1, so the rule must give 1 when n = 1.4 × 1 + ? = 1, so ? = 1 - 4 = -3
- Check with the second term. Put n = 2 in the rule: the answer must be 5.4 × 2 - 3 = 5. ✓
- Write the nth term. 4n - 3
Check. Put n = 2 in 4n - 3: this gives the second term of the sequence. ✓
Watch out. The number in front of n is the difference, but the rule also needs the number that is added (or taken away).
Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. Find the difference: it is the number in front of n. Then find the number that makes the first term right.
- Find the difference between two terms next to each other. It is the same every time, because the sequence is linear.10 - 6 = 4
- Start the rule with 4n, because the difference is 4. The number in front of n is the difference.4n + ?
- Find the number that is added: put n = 1 in the rule. The first term is 6, so the rule must give 6 when n = 1.4 × 1 + ? = 6, so ? = 6 - 4 = 2
- Check with the second term. Put n = 2 in the rule: the answer must be 10.4 × 2 + 2 = 10. ✓
- Write the nth term. 4n + 2
Check. Put n = 2 in 4n + 2: this gives the second term of the sequence. ✓
Watch out. The number in front of n is the difference, but the rule also needs the number that is added (or taken away).
Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. Find the difference: it is the number in front of n. Then find the number that makes the first term right.
- Find the difference between two terms next to each other. It is the same every time. Here it goes down.25 - 30 = -5
- Write the rule as 35 minus 5 times n. The number in front of n is 5, because the sequence goes down by 5.35 - 5n
- Check that n = 1 gives the first term. Put n = 1 in the rule: it must give 30.35 - 5 × 1 = 30. ✓
- Write the nth term. 35 - 5n
Check. Put n = 2 in 35 - 5n: this gives the second term of the sequence. ✓
Watch out. The number in front of n is the difference, but the rule also needs the number that is added (or taken away).
Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.
Method. Put the number in place of the letter, multiply, then add or take away.
- Write the formula. The letter stands for a number we know, so we put it in.3n + 4
- Put 10 in place of n. Multiply first. Multiplication comes before addition and subtraction.3 × 10 = 30
- Add 4. This is the last step of the formula.30 + 4 = 34
- Write the answer. 34
Check. Work it out again: 3 × 10 = 30, then 30 + 4 = 34. ✓
Watch out. Multiply before you add or take away. Do not do the sum from left to right.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. Count the jumps from the first term, multiply by the difference, then add the first term.
- Find the difference. The sequence goes up or down by the same amount each time.difference = 3
- Count the jumps from the first term to term 50. There are 50 - 1 = 49 jumps.50 - 1 = 49 jumps
- Multiply the jumps by the difference. Each jump adds the difference.49 × 3 = 147
- Add the first term. This gives the term.6 + 147 = 153
- Write the answer. 153
Check. Jump back: 153 - 147 = 6, the first term. ✓
Watch out. There are one fewer jumps than the position. The 5th term has 4 jumps.
Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. Solve the equation for n. If n is a whole number that is 1 or more, the number is a term.
- Write the equation. We ask: for which n is the term equal to 100?7n + 2 = 100
- Take 2 away from both sides. This leaves the part with n.7n = 98
- Divide both sides by 7. This gives n, the position of the term.n = 98 ÷ 7 = 14
- Decide. The position must be a whole number, 1 or more.n = 14, a whole number, so 100 is term number 14
- Write the answer. Yes
Check. Put n = 14 in: 7 × 14 + 2 = 100. ✓
Watch out. The position n must be a whole number. A number between two terms is never a term.
Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. Solve the equation for n. If n is a whole number that is 1 or more, the number is a term.
- Write the equation. We ask: for which n is the term equal to 80?6n + 1 = 80
- Take 1 away from both sides. This leaves the part with n.6n = 79
- Divide both sides by 6. This gives n, the position of the term.n = 79 ÷ 6 = 79/6
- Decide. The position must be a whole number, 1 or more.n = 79/6, which is not a whole number, so 80 is not a term
- Write the answer. No
Check. Try the nearest whole numbers: 6 × 13 + 1 and 6 × 14 + 1 are not 80. ✓
Watch out. The position n must be a whole number. A number between two terms is never a term.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.5n - 2 = 73
- Undo the take away 2: add 2 to both sides. Adding the same number to both sides keeps the equation true.5n = 73 + 2 = 75
- Undo the × 5: divide both sides by 5. Dividing both sides by 5 leaves the letter on its own.n = 75 ÷ 5 = 15
- Check: put the answer back into the equation. If both sides match, the answer is right.5 × 15 - 2 = 73 ✓
- Write the answer. n = 15
Check. 5 × 15 - 2 = 73. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. Find the difference: it is the number in front of n. Then find the number that makes the first term right.
- Find the difference between two terms next to each other. It is the same every time, because the sequence is linear.7 - 3 = 4
- Start the rule with 4n, because the difference is 4. The number in front of n is the difference.4n + ?
- Find the number that is added: put n = 1 in the rule. The first term is 3, so the rule must give 3 when n = 1.4 × 1 + ? = 3, so ? = 3 - 4 = -1
- Check with the second term. Put n = 2 in the rule: the answer must be 7.4 × 2 - 1 = 7. ✓
- Write the nth term. 4n - 1
Check. Put n = 2 in 4n - 1: this gives the second term of the sequence. ✓
Watch out. The number in front of n is the difference, but the rule also needs the number that is added (or taken away).
Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. Put n = 1 into Ali’s rule and compare with the first term.
- Write the first term of the sequence. The first term is 5.first term = 5
- Put n = 1 into 4n. This is the first term that Ali’s rule gives.4 × 1 = 4
- Compare the two numbers. They must be the same.4 is not 5
- Write the answer. No
Check. The difference is 4, but the first term is 5, so the rule needs + 1: 4n + 1. ✓
Watch out. Check the rule with n = 1 before you trust it.
Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. List the first few terms of each sequence, then find the first number that appears in both lists.
- Write out the first terms of sequence A, with the rule 3n + 5. Put n = 1, 2, 3 in turn.8, 11, 14, 17, 20, 23, ...
- Write out the first terms of sequence B, with the rule 5n + 1. Put n = 1, 2, 3 in turn.6, 11, 16, 21, 26, ...
- Find the first number that is in both lists. Compare the lists from the start.11 is in both (A when n = 2, B when n = 2)
- Write the answer. 11
Check. 3 × 2 + 5 = 11 and 5 × 2 + 1 = 11. ✓
Watch out. The first number in both lists is the smallest one. Check the next one, 26, does not come before it.