GCSE Maths: Position-to-term rules (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)
Position-to-term rules
A term-to-term rule is slow if you want a term that is far along. To find the 100th term you would need the 99 terms before it. A position-to-term rule (also called the nth term) is much quicker. It lets you find a term directly from its position.
The position of a term is its place in the list: 1st, 2nd, 3rd, and so on. We use the letter n for the position. The rule is a formula with n in it. For example, “nth term = 3n + 2” means “multiply the position by 3, then add 2”. The picture shows this as a machine.
To find a term, replace n by the position number and work it out. With 3n + 2:
| Position n | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 3n + 2 | 3 × 1 + 2 = 5 | 3 × 2 + 2 = 8 | 3 × 3 + 2 = 11 | 3 × 4 + 2 = 14 |
So the sequence is 5, 8, 11, 14, … Use BIDMAS (Chapter 1) when you work it out: brackets and powers first, then multiplying and dividing, then adding and subtracting. The most common mistakes are with powers:
- n² means n × n. When n = 5, n² = 25 (not 10).
- 2n² means 2 × (n × n). When n = 3 this is 2 × 9 = 18. It is not (2 × 3)² = 36.
Terms can be negative. For 10 − 2n the terms are 8, 6, 4, 2, 0, −2, … (n = 6 gives 10 − 12 = −2). Some nth terms are special: n² gives the square numbers 1, 4, 9, 16, …; n³ gives the cube numbers 1, 8, 27, 64, …; 2n gives the powers of 2: 2, 4, 8, 16, … You will see these in Section 11.6.
Look back at 5, 8, 11, 14 from 3n + 2. The term-to-term rule is “add 3”, and the nth term has 3n in it. That is not an accident. The number in front of n is the amount the sequence goes up each time, which is the secret of Section 11.4.
Key ideas
- The position n is the place of the term: the first term has n = 1.
- To find a term, replace n by the position number and work it out using BIDMAS.
- n² = n × n, and 2n² = 2 × (n × n).
- A position-to-term rule jumps straight to any term, such as the 50th or 100th.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = 5n − 3
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 5 × 1 − 3 = 5 − 3 = 2
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 5 × 2 − 3 = 10 − 3 = 7
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 5 × 3 − 3 = 15 − 3 = 12
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 5 × 4 − 3 = 20 − 3 = 17
- Write the terms as a list. 2, 7, 12, 17
Check. The terms should go up by 5 each time (the number in front of n): 5, 5, 5. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = n2 + 4
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 12 + 4 = 1 + 4 = 5
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 22 + 4 = 4 + 4 = 8
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 32 + 4 = 9 + 4 = 13
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 42 + 4 = 16 + 4 = 20
- Replace n by 5 to find the 5th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).5th term = 52 + 4 = 25 + 4 = 29
- Write the terms as a list. 5, 8, 13, 20, 29
Check. The first differences are 3, 5, 7, 9; the second differences are all 2, which is 2 × 1 as expected. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the nth-term rule by the position you want.
- Write the rule. This formula gives the term for any position n.nth term = 2n2 − 1
- The position we want is 6, so replace n by 6. Every n in the formula becomes the same number.6th term = 2 × 62 − 1
- Work it out, doing powers first, then multiplying, then adding or subtracting. This is the BIDMAS order.6th term = 2 × 62 − 1 = 2 × 36 − 1 = 72 − 1 = 71
- Write the answer. The 6th term is 71.
Check. Check on a calculator: the 6th term is 71. ✓
Watch out. Substitute the position number for n. For 3n + 2 with n = 50 you work out 3 × 50 + 2 = 152, you do not multiply anything else by 50.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the nth-term rule by the position you want.
- Write the rule. This formula gives the term for any position n.nth term = 4n + 7
- The position we want is 100, so replace n by 100. Every n in the formula becomes the same number.100th term = 4 × 100 + 7
- Work it out, doing powers first, then multiplying, then adding or subtracting. This is the BIDMAS order.100th term = 4 × 100 + 7 = 400 + 7 = 407
- Write the answer. The 100th term is 407.
Check. The 100th term should be the first term plus 99 steps of 4: 11 + 99 × 4 = 407. ✓
Watch out. Substitute the position number for n. For 3n + 2 with n = 50 you work out 3 × 50 + 2 = 152, you do not multiply anything else by 50.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = 20 − 3n
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 20 − 3 × 1 = 20 − 3 = 17
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 20 − 3 × 2 = 20 − 6 = 14
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 20 − 3 × 3 = 20 − 9 = 11
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 20 − 3 × 4 = 20 − 12 = 8
- Replace n by 5 to find the 5th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).5th term = 20 − 3 × 5 = 20 − 15 = 5
- Replace n by 6 to find the 6th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).6th term = 20 − 3 × 6 = 20 − 18 = 2
- Write the terms as a list. 17, 14, 11, 8, 5, 2
Check. The terms should go up by −3 each time (the number in front of n): −3, −3, −3, −3, −3. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = n2 − 3n + 5
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 12 − 3 × 1 + 5 = 1 − 3 × 1 + 5 = 1 − 3 + 5 = 3
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 22 − 3 × 2 + 5 = 4 − 3 × 2 + 5 = 4 − 6 + 5 = 3
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 32 − 3 × 3 + 5 = 9 − 3 × 3 + 5 = 9 − 9 + 5 = 5
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 42 − 3 × 4 + 5 = 16 − 3 × 4 + 5 = 16 − 12 + 5 = 9
- Replace n by 5 to find the 5th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).5th term = 52 − 3 × 5 + 5 = 25 − 3 × 5 + 5 = 25 − 15 + 5 = 15
- Write the terms as a list. 3, 3, 5, 9, 15
Check. The first differences are 0, 2, 4, 6; the second differences are all 2, which is 2 × 1 as expected. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Watch out
Replace n by the position, then follow BIDMAS. For 2n² with n = 5, square first: 2 × 25 = 50. Do not work out 2 × 5 = 10 and then square it. Also remember that the first term is n = 1, not n = 0.Exercise 11C Position-to-term rules
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. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = 3n + 1
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 3 × 1 + 1 = 3 + 1 = 4
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 3 × 2 + 1 = 6 + 1 = 7
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 3 × 3 + 1 = 9 + 1 = 10
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 3 × 4 + 1 = 12 + 1 = 13
- Write the terms as a list. 4, 7, 10, 13
Check. The terms should go up by 3 each time (the number in front of n): 3, 3, 3. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = 5n − 2
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 5 × 1 − 2 = 5 − 2 = 3
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 5 × 2 − 2 = 10 − 2 = 8
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 5 × 3 − 2 = 15 − 2 = 13
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 5 × 4 − 2 = 20 − 2 = 18
- Write the terms as a list. 3, 8, 13, 18
Check. The terms should go up by 5 each time (the number in front of n): 5, 5, 5. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = 4n + 3
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 4 × 1 + 3 = 4 + 3 = 7
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 4 × 2 + 3 = 8 + 3 = 11
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 4 × 3 + 3 = 12 + 3 = 15
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 4 × 4 + 3 = 16 + 3 = 19
- Write the terms as a list. 7, 11, 15, 19
Check. The terms should go up by 4 each time (the number in front of n): 4, 4, 4. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = 2n − 7
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 2 × 1 − 7 = 2 − 7 = −5
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 2 × 2 − 7 = 4 − 7 = −3
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 2 × 3 − 7 = 6 − 7 = −1
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 2 × 4 − 7 = 8 − 7 = 1
- Write the terms as a list. −5, −3, −1, 1
Check. The terms should go up by 2 each time (the number in front of n): 2, 2, 2. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = 10n + 5
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 10 × 1 + 5 = 10 + 5 = 15
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 10 × 2 + 5 = 20 + 5 = 25
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 10 × 3 + 5 = 30 + 5 = 35
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 10 × 4 + 5 = 40 + 5 = 45
- Write the terms as a list. 15, 25, 35, 45
Check. The terms should go up by 10 each time (the number in front of n): 10, 10, 10. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = 6n − 1
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 6 × 1 − 1 = 6 − 1 = 5
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 6 × 2 − 1 = 12 − 1 = 11
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 6 × 3 − 1 = 18 − 1 = 17
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 6 × 4 − 1 = 24 − 1 = 23
- Write the terms as a list. 5, 11, 17, 23
Check. The terms should go up by 6 each time (the number in front of n): 6, 6, 6. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = 20 − 3n
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 20 − 3 × 1 = 20 − 3 = 17
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 20 − 3 × 2 = 20 − 6 = 14
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 20 − 3 × 3 = 20 − 9 = 11
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 20 − 3 × 4 = 20 − 12 = 8
- Replace n by 5 to find the 5th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).5th term = 20 − 3 × 5 = 20 − 15 = 5
- Write the terms as a list. 17, 14, 11, 8, 5
Check. The terms should go up by −3 each time (the number in front of n): −3, −3, −3, −3. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = 12 − 2n
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 12 − 2 × 1 = 12 − 2 = 10
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 12 − 2 × 2 = 12 − 4 = 8
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 12 − 2 × 3 = 12 − 6 = 6
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 12 − 2 × 4 = 12 − 8 = 4
- Replace n by 5 to find the 5th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).5th term = 12 − 2 × 5 = 12 − 10 = 2
- Write the terms as a list. 10, 8, 6, 4, 2
Check. The terms should go up by −2 each time (the number in front of n): −2, −2, −2, −2. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = 50 − 10n
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 50 − 10 × 1 = 50 − 10 = 40
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 50 − 10 × 2 = 50 − 20 = 30
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 50 − 10 × 3 = 50 − 30 = 20
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 50 − 10 × 4 = 50 − 40 = 10
- Replace n by 5 to find the 5th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).5th term = 50 − 10 × 5 = 50 − 50 = 0
- Write the terms as a list. 40, 30, 20, 10, 0
Check. The terms should go up by −10 each time (the number in front of n): −10, −10, −10, −10. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = 1 − n
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 1 − 1 = 0
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 1 − 2 = −1
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 1 − 3 = −2
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 1 − 4 = −3
- Replace n by 5 to find the 5th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).5th term = 1 − 5 = −4
- Write the terms as a list. 0, −1, −2, −3, −4
Check. The terms should go up by −1 each time (the number in front of n): −1, −1, −1, −1. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = n2 + 1
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 12 + 1 = 1 + 1 = 2
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 22 + 1 = 4 + 1 = 5
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 32 + 1 = 9 + 1 = 10
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 42 + 1 = 16 + 1 = 17
- Write the terms as a list. 2, 5, 10, 17
Check. The first differences are 3, 5, 7; the second differences are all 2, which is 2 × 1 as expected. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = n2 − 1
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 12 − 1 = 1 − 1 = 0
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 22 − 1 = 4 − 1 = 3
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 32 − 1 = 9 − 1 = 8
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 42 − 1 = 16 − 1 = 15
- Write the terms as a list. 0, 3, 8, 15
Check. The first differences are 3, 5, 7; the second differences are all 2, which is 2 × 1 as expected. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = 2n2
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 2 × 12 = 2 × 1 = 2
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 2 × 22 = 2 × 4 = 8
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 2 × 32 = 2 × 9 = 18
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 2 × 42 = 2 × 16 = 32
- Write the terms as a list. 2, 8, 18, 32
Check. The first differences are 6, 10, 14; the second differences are all 4, which is 2 × 2 as expected. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = 3n2 + 2
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 3 × 12 + 2 = 3 × 1 + 2 = 3 + 2 = 5
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 3 × 22 + 2 = 3 × 4 + 2 = 12 + 2 = 14
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 3 × 32 + 2 = 3 × 9 + 2 = 27 + 2 = 29
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 3 × 42 + 2 = 3 × 16 + 2 = 48 + 2 = 50
- Write the terms as a list. 5, 14, 29, 50
Check. The first differences are 9, 15, 21; the second differences are all 6, which is 2 × 3 as expected. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = n3
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 13 = 1
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 23 = 8
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 33 = 27
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 43 = 64
- Write the terms as a list. 1, 8, 27, 64
Check. Re-do the last one on a calculator: the 4th term should be 64. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = 3n
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 31 = 3
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 32 = 9
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 33 = 27
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 34 = 81
- Write the terms as a list. 3, 9, 27, 81
Check. Re-do the last one on a calculator: the 4th term should be 81. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the nth-term rule by the position you want.
- Write the rule. This formula gives the term for any position n.nth term = 4n + 7
- The position we want is 100, so replace n by 100. Every n in the formula becomes the same number.100th term = 4 × 100 + 7
- Work it out, doing powers first, then multiplying, then adding or subtracting. This is the BIDMAS order.100th term = 4 × 100 + 7 = 400 + 7 = 407
- Write the answer. The 100th term is 407.
Check. The 100th term should be the first term plus 99 steps of 4: 11 + 99 × 4 = 407. ✓
Watch out. Substitute the position number for n. For 3n + 2 with n = 50 you work out 3 × 50 + 2 = 152, you do not multiply anything else by 50.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the nth-term rule by the position you want.
- Write the rule. This formula gives the term for any position n.nth term = 3n − 5
- The position we want is 50, so replace n by 50. Every n in the formula becomes the same number.50th term = 3 × 50 − 5
- Work it out, doing powers first, then multiplying, then adding or subtracting. This is the BIDMAS order.50th term = 3 × 50 − 5 = 150 − 5 = 145
- Write the answer. The 50th term is 145.
Check. The 50th term should be the first term plus 49 steps of 3: −2 + 49 × 3 = 145. ✓
Watch out. Substitute the position number for n. For 3n + 2 with n = 50 you work out 3 × 50 + 2 = 152, you do not multiply anything else by 50.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the nth-term rule by the position you want.
- Write the rule. This formula gives the term for any position n.nth term = 9n + 4
- The position we want is 20, so replace n by 20. Every n in the formula becomes the same number.20th term = 9 × 20 + 4
- Work it out, doing powers first, then multiplying, then adding or subtracting. This is the BIDMAS order.20th term = 9 × 20 + 4 = 180 + 4 = 184
- Write the answer. The 20th term is 184.
Check. The 20th term should be the first term plus 19 steps of 9: 13 + 19 × 9 = 184. ✓
Watch out. Substitute the position number for n. For 3n + 2 with n = 50 you work out 3 × 50 + 2 = 152, you do not multiply anything else by 50.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the nth-term rule by the position you want.
- Write the rule. This formula gives the term for any position n.nth term = 100 − n
- The position we want is 60, so replace n by 60. Every n in the formula becomes the same number.60th term = 100 − 60
- Work it out, doing powers first, then multiplying, then adding or subtracting. This is the BIDMAS order.60th term = 100 − 60 = 40
- Write the answer. The 60th term is 40.
Check. The 60th term should be the first term plus 59 steps of −1: 99 + 59 × (−1) = 40. ✓
Watch out. Substitute the position number for n. For 3n + 2 with n = 50 you work out 3 × 50 + 2 = 152, you do not multiply anything else by 50.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the nth-term rule by the position you want.
- Write the rule. This formula gives the term for any position n.nth term = n2 + 3
- The position we want is 12, so replace n by 12. Every n in the formula becomes the same number.12th term = 122 + 3
- Work it out, doing powers first, then multiplying, then adding or subtracting. This is the BIDMAS order.12th term = 122 + 3 = 144 + 3 = 147
- Write the answer. The 12th term is 147.
Check. Check on a calculator: the 12th term is 147. ✓
Watch out. Substitute the position number for n. For 3n + 2 with n = 50 you work out 3 × 50 + 2 = 152, you do not multiply anything else by 50.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the nth-term rule by the position you want.
- Write the rule. This formula gives the term for any position n.nth term = 2n2 + 1
- The position we want is 10, so replace n by 10. Every n in the formula becomes the same number.10th term = 2 × 102 + 1
- Work it out, doing powers first, then multiplying, then adding or subtracting. This is the BIDMAS order.10th term = 2 × 102 + 1 = 2 × 100 + 1 = 200 + 1 = 201
- Write the answer. The 10th term is 201.
Check. Check on a calculator: the 10th term is 201. ✓
Watch out. Substitute the position number for n. For 3n + 2 with n = 50 you work out 3 × 50 + 2 = 152, you do not multiply anything else by 50.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = 3n − 2
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 3 × 1 − 2 = 3 − 2 = 1
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 3 × 2 − 2 = 6 − 2 = 4
- Replace n by 5 to find the 5th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).5th term = 3 × 5 − 2 = 15 − 2 = 13
- Replace n by 10 to find the 10th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).10th term = 3 × 10 − 2 = 30 − 2 = 28
- Write the terms as a list. 1, 4, 13, 28
Check. Re-do the last one on a calculator: the 10th term should be 28. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the nth-term rule by the position you want.
- Write the rule. This formula gives the term for any position n.nth term = 2n + 3
- The position we want is 7, so replace n by 7. Every n in the formula becomes the same number.7th term = 2 × 7 + 3
- Work it out, doing powers first, then multiplying, then adding or subtracting. This is the BIDMAS order.7th term = 2 × 7 + 3 = 14 + 3 = 17
- Write the answer. The 7th term is 17.
Check. The 7th term should be the first term plus 6 steps of 2: 5 + 6 × 2 = 17. ✓
Watch out. Substitute the position number for n. For 3n + 2 with n = 50 you work out 3 × 50 + 2 = 152, you do not multiply anything else by 50.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the nth-term rule by the position you want.
- Write the rule. This formula gives the term for any position n.nth term = 2n + 3
- The position we want is 15, so replace n by 15. Every n in the formula becomes the same number.15th term = 2 × 15 + 3
- Work it out, doing powers first, then multiplying, then adding or subtracting. This is the BIDMAS order.15th term = 2 × 15 + 3 = 30 + 3 = 33
- Write the answer. The 15th term is 33.
Check. The 15th term should be the first term plus 14 steps of 2: 5 + 14 × 2 = 33. ✓
Watch out. Substitute the position number for n. For 3n + 2 with n = 50 you work out 3 × 50 + 2 = 152, you do not multiply anything else by 50.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the nth-term rule by the position you want.
- Write the rule. This formula gives the term for any position n.nth term = 2n + 3
- The position we want is 40, so replace n by 40. Every n in the formula becomes the same number.40th term = 2 × 40 + 3
- Work it out, doing powers first, then multiplying, then adding or subtracting. This is the BIDMAS order.40th term = 2 × 40 + 3 = 80 + 3 = 83
- Write the answer. The 40th term is 83.
Check. The 40th term should be the first term plus 39 steps of 2: 5 + 39 × 2 = 83. ✓
Watch out. Substitute the position number for n. For 3n + 2 with n = 50 you work out 3 × 50 + 2 = 152, you do not multiply anything else by 50.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the nth-term rule by the position you want.
- Write the rule. This formula gives the term for any position n.nth term = 15n + 25
- The position we want is 12, so replace n by 12. Every n in the formula becomes the same number.12th term = 15 × 12 + 25
- Work it out, doing powers first, then multiplying, then adding or subtracting. This is the BIDMAS order.12th term = 15 × 12 + 25 = 180 + 25 = 205
- Write the answer. The 12th term is 205.
Check. The 12th term should be the first term plus 11 steps of 15: 40 + 11 × 15 = 205. ✓
Watch out. Substitute the position number for n. For 3n + 2 with n = 50 you work out 3 × 50 + 2 = 152, you do not multiply anything else by 50.
Method. Check what 2n² means, then work it out in the right order.
- Say what 2n² means. The square only applies to n, not to the 2. So 2n² = 2 × (n × n).2n² = 2 × n × n
- Work it out for n = 5 in BIDMAS order. Powers come before multiplying.5² = 25, then 2 × 25 = 50
- Find the mistake. Zoe multiplied first and then squared, which would be (2 × 5)².She worked out (2n)² = 100 instead of 2n² = 50.
Check. Try n = 2: 2n² = 2 × 4 = 8, and (2n)² = 4² = 16, which is different. ✓
Watch out. A power belongs to the letter or bracket just before it.
Method. Test each rule with n = 1 and n = 2 and compare with the sequence 5, 8, …
- Test A, 5n. Replace n by 1, then 2.5 × 1 = 5 ✓, 5 × 2 = 10, but the 2nd term is 8 ✗
- Test B, n + 4. 1 + 4 = 5 ✓, 2 + 4 = 6, but the 2nd term is 8 ✗
- Test C, 3n + 2. 3 × 1 + 2 = 5 ✓, 3 × 2 + 2 = 8 ✓, 3 × 3 + 2 = 11 ✓
- Test D, 3n + 5 for completeness. 3 × 1 + 5 = 8, but the 1st term is 5 ✗
Check. 3n + 2 with n = 4 gives 14, the 4th term. ✓
Watch out. Test at least two positions: a wrong rule can match the first term by luck.
Method. Work out both 10th terms.
- Find the 10th term of n². 10² = 100
- Find the 10th term of 11n. 11 × 10 = 110
- Compare. 110 > 100, so 11n is bigger.
Check. At n = 11 both are 121, so the two sequences meet there; before that 11n is bigger. ✓
Method. Work out both 12th terms.
- Find the 12th term of n². 12² = 144
- Find the 12th term of 11n. 11 × 12 = 132
- Compare. 144 > 132, so n² is bigger now.
Check. The squares overtake 11n at n = 11 where both are 121. ✓
Watch out. A squared term eventually grows faster than a multiple of n.
Method. Follow the machine in order.
- Multiply by 4. n × 4 = 4n
- Subtract 3. 4n − 3
Check. With n = 2: 2 × 4 = 8, 8 − 3 = 5, and 4 × 2 − 3 = 5. ✓
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the nth-term rule by the position you want.
- Write the rule. This formula gives the term for any position n.nth term = 4n − 3
- The position we want is 6, so replace n by 6. Every n in the formula becomes the same number.6th term = 4 × 6 − 3
- Work it out, doing powers first, then multiplying, then adding or subtracting. This is the BIDMAS order.6th term = 4 × 6 − 3 = 24 − 3 = 21
- Write the answer. The 6th term is 21.
Check. The 6th term should be the first term plus 5 steps of 4: 1 + 5 × 4 = 21. ✓
Watch out. Substitute the position number for n. For 3n + 2 with n = 50 you work out 3 × 50 + 2 = 152, you do not multiply anything else by 50.
Method. Find the two terms, then add them.
- Find the 1st term. Replace n by 1.1st term = 7 × 1 − 4 = 7 − 4 = 3
- Find the 10th term. Replace n by 10.10th term = 7 × 10 − 4 = 70 − 4 = 66
- Add them. The question asks for the sum.3 + 66 = 69
Check. Another way: the 1st and 10th terms are 9 steps of 7 apart: 3 + 9 × 7 = 66. ✓
Watch out. Do not add the two positions; add the two terms.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = n2 − 3n + 5
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 12 − 3 × 1 + 5 = 1 − 3 × 1 + 5 = 1 − 3 + 5 = 3
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 22 − 3 × 2 + 5 = 4 − 3 × 2 + 5 = 4 − 6 + 5 = 3
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 32 − 3 × 3 + 5 = 9 − 3 × 3 + 5 = 9 − 9 + 5 = 5
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 42 − 3 × 4 + 5 = 16 − 3 × 4 + 5 = 16 − 12 + 5 = 9
- Replace n by 5 to find the 5th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).5th term = 52 − 3 × 5 + 5 = 25 − 3 × 5 + 5 = 25 − 15 + 5 = 15
- Write the terms as a list. 3, 3, 5, 9, 15
Check. The first differences are 0, 2, 4, 6; the second differences are all 2, which is 2 × 1 as expected. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = 2n2 + n − 1
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 2 × 12 + 1 − 1 = 2 × 1 + 1 − 1 = 2 + 1 − 1 = 2
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 2 × 22 + 2 − 1 = 2 × 4 + 2 − 1 = 8 + 2 − 1 = 9
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 2 × 32 + 3 − 1 = 2 × 9 + 3 − 1 = 18 + 3 − 1 = 20
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 2 × 42 + 4 − 1 = 2 × 16 + 4 − 1 = 32 + 4 − 1 = 35
- Replace n by 5 to find the 5th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).5th term = 2 × 52 + 5 − 1 = 2 × 25 + 5 − 1 = 50 + 5 − 1 = 54
- Write the terms as a list. 2, 9, 20, 35, 54
Check. The first differences are 7, 11, 15, 19; the second differences are all 4, which is 2 × 2 as expected. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = −n2 + 10n
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = −12 + 10 × 1 = −1 + 10 × 1 = −1 + 10 = 9
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = −22 + 10 × 2 = −4 + 10 × 2 = −4 + 20 = 16
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = −32 + 10 × 3 = −9 + 10 × 3 = −9 + 30 = 21
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = −42 + 10 × 4 = −16 + 10 × 4 = −16 + 40 = 24
- Replace n by 5 to find the 5th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).5th term = −52 + 10 × 5 = −25 + 10 × 5 = −25 + 50 = 25
- Write the terms as a list. 9, 16, 21, 24, 25
Check. The first differences are 7, 5, 3, 1; the second differences are all −2, which is 2 × −1 as expected. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Key words. The position of a term is its place in the sequence: 1st, 2nd, 3rd and so on. We use the letter n for the position number. The nth term (or position-to-term rule) is a formula that gives the term for any position n.
Method. Replace n in the formula by each position number, and work out the answer.
- Write the rule and say what it means. The letter n stands for the position of the term (1 for the first term, 2 for the second, and so on). To get a term, replace n by the position number.nth term = n2 + n
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 12 + 1 = 1 + 1 = 2
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 22 + 2 = 4 + 2 = 6
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 32 + 3 = 9 + 3 = 12
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 42 + 4 = 16 + 4 = 20
- Replace n by 5 to find the 5th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).5th term = 52 + 5 = 25 + 5 = 30
- Write the terms as a list. 2, 6, 12, 20, 30
Check. The first differences are 4, 6, 8, 10; the second differences are all 2, which is 2 × 1 as expected. ✓
Watch out. n² means n × n, not 2 × n. And the first term is when n = 1, not n = 0.
Method. Work out the terms and compare them.
- Find the 2nd term. Replace n by 2.2nd term = 22 − 6 × 2 + 10 = 4 − 6 × 2 + 10 = 4 − 12 + 10 = 2
- Find the 4th term. Replace n by 4.4th term = 42 − 6 × 4 + 10 = 16 − 6 × 4 + 10 = 16 − 24 + 10 = 2
- Conclude. Both give 2, so they are equal. ✓
- Look for another pair. Work out the 1st, 3rd and 5th terms.1st term = 12 − 6 × 1 + 10 = 1 − 6 × 1 + 10 = 1 − 6 + 10 = 5
5th term = 52 − 6 × 5 + 10 = 25 − 6 × 5 + 10 = 25 − 30 + 10 = 5
Both are 5, so the 1st and 5th terms are equal.
Check. The terms go 5, 2, 1, 2, 5: they are symmetrical about the 3rd term. ✓
Watch out. This happens because the sequence goes down and then back up again.