GCSE Maths: Using the nth term (Foundation)
Part of the GCSE Maths sequences and nth term chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Using the nth term
Once you have an nth term you can do much more than find single terms. This section shows four uses.
1. Is a number in the sequence? A number is a term only if it comes from a whole-number position such as 1, 2, 3, … You cannot have the 4.5th term. So set the nth term equal to the number and solve for n. If n is a positive whole number, the number is in the sequence (and n is its position). If not, it is not. For example, is 62 in the sequence 4n − 3? Solving 4n − 3 = 62 gives 4n = 65 and n = 16.25, which is not a whole number, so no.
2. Which term is it? The same method gives the position. If 5n + 2 = 97 then 5n = 95, so n = 19: 97 is the 19th term.
3. The first term bigger than a number. Write an inequality, for example 7n − 4 > 200. Solve it as an equation (7n > 204, so n > 29.14…) and then pick the next whole number up, which is 30. The 30th term is 206, and the 29th is 199, which is too small. In the same way, for the largest term below a number you round down.
4. The nth term of a pattern. To find a formula for a pattern of matchsticks or tiles, count each pattern to make a sequence, find its nth term, and then check that the formula matches how the picture is built. For the rows of squares, the counts 4, 7, 10, 13 have common difference 3 and nth term 3n + 1. In the picture, each square has 3 matchsticks of its own and there is 1 extra at the start.
Key ideas
- A number is in the sequence only if solving “nth term = number” gives a positive whole number n.
- To find the first term above a value, solve an inequality and round n up; for the largest term below a value, round down.
- For a pattern, count to make a sequence, find the nth term, then explain it from the picture.
- Higher: if two terms of an arithmetic sequence are known, divide the change in the terms by the change in the positions to get d.
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.
- 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 = 62? - 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 4.4n − 3 = 62
4n = 62 + 3 = 65 - Divide by 4. The inverse of multiplying by 4 is dividing by 4.n = 65 ÷ 4 = 16.25
- Decide. 16.25 is not a whole number, so there is no such position.No: 62 is not in the sequence.
Check. The terms either side are the 16th term, 61, and the 17th term, 65. 62 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.
- 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. - 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 + 2
Is there a whole number n with 5n + 2 = 97? - 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 5.5n + 2 = 97
5n = 97 − 2 = 95 - Divide by 5. The inverse of multiplying by 5 is dividing by 5.n = 95 ÷ 5 = 19
- Decide. 19 is a whole number and it is at least 1, so it is a real position.Yes: 97 is the 19th term.
Check. Substitute n = 19: 5 × 19 + 2 = 95 + 2 = 97, which is 97. ✓
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.
- Write what we want as an inequality. We want the first term that is greater than 200. The terms must be greater than 200.7n − 4 > 200
- Solve it like an equation. Add 4 on both sides, then divide both sides by 7.7n > 200 + 4 = 204
n > 204 ÷ 7 = 2047 - Choose the smallest whole number that works. Positions must be whole numbers. n has to be bigger than 2047, so take the next whole number up.n = 30
- Work out the term and test the one before it. The term in position n must be greater than 200, and the term before it must not be.30th term = 7 × 30 − 4 = 210 − 4 = 206
29th term = 7 × 29 − 4 = 203 − 4 = 199 (not greater than 200)
Check. 206 is greater than 200 and 199 is not. ✓
Watch out. Do not round n to the nearest whole number. You need the first whole number that makes the term big (or small) enough, which means rounding up.
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. Solve an inequality to find how far along the sequence you can go, then pick the largest whole number that works.
- Write what we want as an inequality. We want terms that are less than 500.8n + 3 < 500
- Solve it like an equation. Subtract 3 on both sides, then divide both sides by 8.8n < 497
n < 497 ÷ 8 = 62.125 - Choose the largest whole number that works. n must be a whole number less than 62.125.n = 62
- Work out the term and test the next one. The term in position n must be less than 500, and the next one must not be.62nd term = 8 × 62 + 3 = 496 + 3 = 499
63rd term = 8 × 63 + 3 = 504 + 3 = 507 (too big)
Check. 499 < 500 but 507 > 500 or equal. ✓
Watch out. Round down here, because the term must stay below the limit.
Find the nth term. How many matchsticks are needed for pattern 30?
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.
- 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 - 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 - 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 - 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 3n 3 6 9 Term − 3n 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 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.
- Replace n by 30. Pattern 30 is the one in position 30.Pattern 30: 3 × 30 + 1 = 90 + 1 = 91
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.
Watch out
Do not round n to the nearest whole number. If 7n − 4 > 200 gives n > 29.14, then n = 29 does not work (the 29th term is 199). You need the next whole number up, 30. And a number is only in the sequence if n comes out as an exact whole number.Exercise 11E Using the nth term
Non-calculatorKey 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.
- 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? - 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 - Divide by 3. The inverse of multiplying by 3 is dividing by 3.n = 30 ÷ 3 = 10
- 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.
- 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? - 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 - Divide by 3. The inverse of multiplying by 3 is dividing by 3.n = 49 ÷ 3 = 493
- 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.
- 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? - 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 - Divide by 3. The inverse of multiplying by 3 is dividing by 3.n = 99 ÷ 3 = 33
- 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.
- 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? - 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 - Divide by 3. The inverse of multiplying by 3 is dividing by 3.n = 150 ÷ 3 = 50
- 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.
- 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 − 2
Is there a whole number n with 5n − 2 = 48? - Set the nth term equal to the number and solve for n. Add 2 on both sides to get the n term on its own, then divide by 5.5n − 2 = 48
5n = 48 + 2 = 50 - Divide by 5. The inverse of multiplying by 5 is dividing by 5.n = 50 ÷ 5 = 10
- Decide. 10 is a whole number and it is at least 1, so it is a real position.Yes: 48 is the 10th term.
Check. Substitute n = 10: 5 × 10 − 2 = 50 − 2 = 48, which is 48. ✓
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.
- 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 − 2
Is there a whole number n with 5n − 2 = 93? - Set the nth term equal to the number and solve for n. Add 2 on both sides to get the n term on its own, then divide by 5.5n − 2 = 93
5n = 93 + 2 = 95 - Divide by 5. The inverse of multiplying by 5 is dividing by 5.n = 95 ÷ 5 = 19
- Decide. 19 is a whole number and it is at least 1, so it is a real position.Yes: 93 is the 19th term.
Check. Substitute n = 19: 5 × 19 − 2 = 95 − 2 = 93, which is 93. ✓
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.
- 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 − 2
Is there a whole number n with 5n − 2 = 198? - Set the nth term equal to the number and solve for n. Add 2 on both sides to get the n term on its own, then divide by 5.5n − 2 = 198
5n = 198 + 2 = 200 - Divide by 5. The inverse of multiplying by 5 is dividing by 5.n = 200 ÷ 5 = 40
- Decide. 40 is a whole number and it is at least 1, so it is a real position.Yes: 198 is the 40th term.
Check. Substitute n = 40: 5 × 40 − 2 = 200 − 2 = 198, which is 198. ✓
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.
- 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 - 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 - 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 - 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 4n.Term 7 11 15 19 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, 7 − 4 = 3. - 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? - 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 - Divide by 4. The inverse of multiplying by 4 is dividing by 4.n = 97 ÷ 4 = 24.25
- 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.
- 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 - 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 - 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 - 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 6n.Term 3 9 15 21 6n 6 12 18 24 Term − 6n −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 = 6n − 3
Shortcut for later: the adjustment −3 is the term before the first term, 3 − 6 = −3. - 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? - 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 - Divide by 6. The inverse of multiplying by 6 is dividing by 6.n = 102 ÷ 6 = 17
- 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.
- 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 - 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 - 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 - 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 3n.Term 2 5 8 11 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, 2 − 3 = −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 = 61? - 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 - Divide by 3. The inverse of multiplying by 3 is dividing by 3.n = 62 ÷ 3 = 623
- 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.
- 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 - 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 - 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 - 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 5n.Term 10 15 20 25 5n 5 10 15 20 Term − 5n 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 = 5n + 5
Shortcut for later: the adjustment 5 is the term before the first term, 10 − 5 = 5. - 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? - 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 - Divide by 5. The inverse of multiplying by 5 is dividing by 5.n = 82 ÷ 5 = 16.4
- 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.
- 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? - 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 - Divide by (−3). The inverse of multiplying by −3 is dividing by −3.n = −30 ÷ (−3) = 10
- 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.
- 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? - 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 - Divide by (−3). The inverse of multiplying by −3 is dividing by −3.n = −31 ÷ (−3) = 313
- 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.
- 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? - 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 - Divide by (−3). The inverse of multiplying by −3 is dividing by −3.n = −18 ÷ (−3) = 6
- 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.
- 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? - 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 - Divide by (−3). The inverse of multiplying by −3 is dividing by −3.n = −20 ÷ (−3) = 203
- 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.
- 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
- Solve it like an equation. Add 4 on both sides, then divide both sides by 7.7n > 100 + 4 = 104
n > 104 ÷ 7 = 1047 - 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
- 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.
- 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
- 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 - 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
- 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.
- 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
- Solve it like an equation. Subtract 2 on both sides, then divide both sides by 9.9n > 500 − 2 = 498
n > 498 ÷ 9 = 1663 - 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
- 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.
- 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
- 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 - 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
- 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.
- 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
- 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 - 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
- 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.
- 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
- 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 - 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
- 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.
- 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
- 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 - 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
- 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.
- 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 - 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 - 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 - 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 3n 3 6 9 Term − 3n 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 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.
- 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 - 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 - 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 - 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 3n 3 6 9 Term − 3n 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 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.
- 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.
- 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 - 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 - 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 - 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 2n 2 4 6 Term − 2n 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. - 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.
- 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 - 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 - 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 - 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 2n 2 4 6 Term − 2n 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. - 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.
- 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 - 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.
- 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 - 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 - 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 - 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 4n.Term 5 9 13 4n 4 8 12 Term − 4n 1 1 1 - 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. - 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.
- 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 - 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 - 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 - 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 4n.Term 5 9 13 4n 4 8 12 Term − 4n 1 1 1 - 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. - 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.
- 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.
- 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 - 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 - 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 - 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 4n.Term 5 9 13 4n 4 8 12 Term − 4n 1 1 1 - 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. - 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.
- 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 - 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.
- 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 - 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 - 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 - 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 add 4 to 4n.Term 8 12 16 4n 4 8 12 Term − 4n 4 4 4 - 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. - 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.
- 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 - 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 - 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 - 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 add 4 to 4n.Term 8 12 16 4n 4 8 12 Term − 4n 4 4 4 - 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. - 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.
- 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.
- 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? - 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 - Divide by 2. The inverse of multiplying by 2 is dividing by 2.n = 44 ÷ 2 = 22
- 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.
- 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? - 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 - Divide by 2. The inverse of multiplying by 2 is dividing by 2.n = 47 ÷ 2 = 23.5
- 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.
- 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? - 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 - Divide by 4. The inverse of multiplying by 4 is dividing by 4.n = 98 ÷ 4 = 24.5
- 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.
- 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? - 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 - Divide by 4. The inverse of multiplying by 4 is dividing by 4.n = 100 ÷ 4 = 25
- 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. 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. Solve an inequality to find how far along the sequence you can go, then pick the largest whole number that works.
- Write what we want as an inequality. We want terms that are less than 300.11n + 5 < 300
- Solve it like an equation. Subtract 5 on both sides, then divide both sides by 11.11n < 295
n < 295 ÷ 11 = 29511 - Choose the largest whole number that works. n must be a whole number less than 29511.n = 26
- Work out the term and test the next one. The term in position n must be less than 300, and the next one must not be.26th term = 11 × 26 + 5 = 286 + 5 = 291
27th term = 11 × 27 + 5 = 297 + 5 = 302 (too big)
Check. 291 < 300 but 302 > 300 or equal. ✓
Watch out. Round down here, because the term must stay below the limit.
Method. Rewrite 6n − 2 as a multiple of 3 plus a remainder.
- Look at 6n. 6 is a multiple of 3, so 6n is a multiple of 3 for every whole number n.6n = 3 × (2n)
- Write the term another way. We can split off a 3 so that what is left over is small.6n − 2 = 6n − 3 + 1 = 3(2n − 1) + 1
- Explain. 3(2n − 1) is a multiple of 3, so every term is 1 more than a multiple of 3.Dividing any term by 3 leaves a remainder of 1, so no term is a multiple of 3.
Check. The terms are 4, 10, 16, 22, …: each divided by 3 leaves remainder 1 (4 = 3 + 1, 10 = 9 + 1, 16 = 15 + 1). ✓
Watch out. Examples are not proof. Use the algebra to show it works for every term.
Method. Use what you know about even and odd numbers.
- Look at 2n. 2 times any whole number is even.2n is even for every whole number n
- Add 1. An even number plus 1 is odd.2n + 1 is odd
- Apply to 100. Every term is odd, but 100 is even.100 cannot be a term, so no pattern uses exactly 100 matchsticks.
Check. Solve 2n + 1 = 100: 2n = 99 and n = 49.5, which is not a whole number. ✓
Watch out. Give the reason (even/odd), not just a check of a few numbers.