GCSE Maths: Special sequences (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)
Special sequences
Some sequences turn up so often that they have names. You should know them well: you will see them in non-calculator exams. Here they are, with the nth term of each.
| Square numbers | 1 | 4 | 9 | 16 | 25 | 36 | 49 | … |
|---|---|---|---|---|---|---|---|---|
| Cube numbers | 1 | 8 | 27 | 64 | 125 | 216 | 343 | … |
| Triangular numbers | 1 | 3 | 6 | 10 | 15 | 21 | 28 | … |
| Powers of 2 | 2 | 4 | 8 | 16 | 32 | 64 | 128 | … |
| Powers of 10 | 10 | 100 | 1000 | 10 000 | … |
Square numbers. The nth term is n² = n × n. They are the numbers of dots in a square pattern. Cube numbers. The nth term is n³ = n × n × n. They count the small cubes in a bigger cube.
Triangular numbers. These count the dots in triangles with rows of 1, 2, 3, … dots: 1, 3, 6, 10, 15, … The nth term is n(n + 1) ÷ 2. Why? Put two identical triangles together and you get a rectangle that is n dots by (n + 1) dots. One triangle is half of the rectangle. Here is the idea for n = 4: two triangles of 10 dots make a 4 by 5 rectangle of 20 dots.
Powers of 2 and 10. These are made by multiplying by 2 (or by 10) each time. They are examples of geometric sequences (Section 11.7). The nth term is 2n, which means n twos multiplied together, so 25 = 2 × 2 × 2 × 2 × 2 = 32.
Fibonacci-type sequences. In a Fibonacci-type sequence you make each term by adding the two terms before it. The famous Fibonacci sequence starts 1, 1, 2, 3, 5, 8, 13, 21, … Other Fibonacci-type sequences start with different numbers, for example 2, 5, 7, 12, 19, … (2 + 5 = 7, 5 + 7 = 12, 7 + 12 = 19). To find a missing term, add the two terms before it, or work backwards by subtracting.
How to recognise them. Look at the differences and ratios. Differences 3, 5, 7, 9 (odd numbers going up by 2) mean square numbers. Differences 2, 3, 4, 5 mean triangular numbers. If each term is double the last, they are powers of 2. If each term is the sum of the two before, it is Fibonacci-type.
Key ideas
- Squares: 1, 4, 9, 16, 25, … nth term n²
- Cubes: 1, 8, 27, 64, 125, … nth term n³
- Triangular: 1, 3, 6, 10, 15, … nth term n(n + 1) ÷ 2
- Powers of 2: 2, 4, 8, 16, 32, … nth term 2n
- Fibonacci-type: each term is the sum of the two before it.
- Learn the first ten squares, the first five cubes and the first eight triangular numbers.
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. Use the nth-term rule for triangular numbers.
- Remember what triangular numbers are. A triangular number is the number of dots in a triangle with rows of 1, 2, 3, … dots. The nth triangular number is 1 + 2 + … + n. Two copies of the same triangle fit together to make a rectangle n by (n + 1), so one copy is half of that: n(n + 1) ÷ 2.nth term = n(n + 1)2
- Replace n by 15. We want the term in position 15, so use 15 in place of every n.15th term = 15 × (15 + 1) ÷ 2
- Work it out. Follow the order of operations: brackets, powers, multiply and divide, then add and subtract.15th term = 15 × (15 + 1) ÷ 2 = 15 × 16 ÷ 2 = 240 ÷ 2 = 120
Check. Add the numbers 1 to 15 directly: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 = 120. ✓
Watch out. Multiply n by (n + 1) first, and only then divide by 2.
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. Recognise the sequence by its differences or ratios and by comparing with the special sequences you know.
- Find the differences between neighbouring terms. This tells us what kind of sequence it is.8 − 1 = 7
27 − 8 = 19
64 − 27 = 37
These are 7, 19, 37: going up by a bigger amount each time (the differences 7, 19, 37, … are not simple). - Compare with the special sequences you know. It helps to know these by heart: squares 1, 4, 9, 16, 25; cubes 1, 8, 27, 64, 125; triangular numbers 1, 3, 6, 10, 15; powers of 2: 2, 4, 8, 16; powers of 10: 10, 100, 1000.The cube numbers are 1, 8, 27, 64, 125, 216, 343, …
Our sequence 1, 8, 27, 64 appears in this list. - Use the rule for the cube numbers to find the next 2 terms. The rule is nth term = n3.5th term of the cube numbers: 53 = 125
6th term of the cube numbers: 63 = 216
Check. The next terms 125, 216 carry on the same pattern as 27, 64. ✓
Watch out. A sequence that looks like squares might be something else. Check at least three terms, and use the differences.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. A Fibonacci-type sequence is built by adding the previous two terms.
- Write the first two terms. A Fibonacci-type sequence needs two starting terms.2, 3
- Add the last two terms to make each new term. In a Fibonacci-type sequence every term is the sum of the two terms before it.Term 3 = 2 + 3 = 5
Term 4 = 3 + 5 = 8
Term 5 = 5 + 8 = 13
Term 6 = 8 + 13 = 21
Term 7 = 13 + 21 = 34
Term 8 = 21 + 34 = 55 - Write the sequence. 2, 3, 5, 8, 13, 21, 34, 55
Check. Check the last term: 21 + 34 = 55. ✓
Watch out. Always add the last two terms, not the first two.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Every term is the sum of the previous two. To find a missing term, add the two before it, or subtract to work backwards.
- Use the rule: each term is the sum of the two terms before it. Look for a gap where you know the two terms before it (add them) or the two terms after it (subtract).Terms: 3, 5, ?, 13, ?
- Find term 3. Term 3 is the sum of the two terms before it.3 + 5 = 8
- Find term 5. Term 5 is the sum of the two terms before it.8 + 13 = 21
- Write the complete sequence. 3, 5, 8, 13, 21
Check. Check every term is the sum of the two before it: 3 + 5 = 8, 5 + 8 = 13, 8 + 13 = 21. ✓
Watch out. When working backwards, subtract: the missing term is the later term minus its neighbour.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Every term is the sum of the previous two. To find a missing term, add the two before it, or subtract to work backwards.
- Use the rule: each term is the sum of the two terms before it. Look for a gap where you know the two terms before it (add them) or the two terms after it (subtract).Terms: ?, ?, 7, 11, 18
- Find term 2. Term 4 is the sum of terms 2 and 3, so term 2 = term 4 − term 3. We are working backwards.11 − 7 = 4
- Find term 1. Term 3 is the sum of terms 1 and 2, so term 1 = term 3 − term 2. We are working backwards.7 − 4 = 3
- Write the complete sequence. 3, 4, 7, 11, 18
Check. Check every term is the sum of the two before it: 3 + 4 = 7, 4 + 7 = 11, 7 + 11 = 18. ✓
Watch out. When working backwards, subtract: the missing term is the later term minus its neighbour.
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. Work backwards from the term to the position.
- Remember the rule for square numbers. A square number is a whole number multiplied by itself. It is the number of small squares in a bigger square with that side length.nth term = n2
- Use the square root. The inverse (undo) of squaring is the square root. We need the number that multiplies by itself to give 144.√144 = 12, because 12 × 12 = 144
- Write the answer. The position is the value of n.144 is the 12th term of the square numbers.
Check. Substitute n = 12: 122 = 144. ✓
Watch out. The answer is the position (n), not the term. Say which one you are giving.
Watch out
n² is n × n, not 2 × n. The square numbers go 1, 4, 9, 16, but 2n goes 2, 4, 6, 8. In the same way, 2n (2, 4, 8, 16, …) is not the same as 2n or n². They only look alike at the start: all three have 4 as the second term.Exercise 11F Special sequences
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. Use the nth-term rule for square numbers.
- Remember what square numbers are. A square number is a whole number multiplied by itself. It is the number of small squares in a bigger square with that side length.nth term = n2
- Replace n by 9. We want the term in position 9, so use 9 in place of every n.9th term = 92
- Work it out. Follow the order of operations: brackets, powers, multiply and divide, then add and subtract.9th term = 92 = 81
Check. The sequence 1, 4, 9, 16, … matches the start of the square numbers, and the 9th term follows the same rule. ✓
Watch out. Do not confuse n² with 2n: 5² = 25 but 2 × 5 = 10.
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. Use the nth-term rule for cube numbers.
- Remember what cube numbers are. A cube number is a whole number multiplied by itself, and then by itself again. It is the number of small cubes in a bigger cube with that side length.nth term = n3
- Replace n by 4. We want the term in position 4, so use 4 in place of every n.4th term = 43
- Work it out. Follow the order of operations: brackets, powers, multiply and divide, then add and subtract.4th term = 43 = 64
Check. The sequence 1, 8, 27, 64, … matches the start of the cube numbers, and the 4th term follows the same rule. ✓
Watch out. Do not confuse n² with 2n: 5² = 25 but 2 × 5 = 10.
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. Use the nth-term rule for triangular numbers.
- Remember what triangular numbers are. A triangular number is the number of dots in a triangle with rows of 1, 2, 3, … dots. The nth triangular number is 1 + 2 + … + n. Two copies of the same triangle fit together to make a rectangle n by (n + 1), so one copy is half of that: n(n + 1) ÷ 2.nth term = n(n + 1)2
- Replace n by 7. We want the term in position 7, so use 7 in place of every n.7th term = 7 × (7 + 1) ÷ 2
- Work it out. Follow the order of operations: brackets, powers, multiply and divide, then add and subtract.7th term = 7 × (7 + 1) ÷ 2 = 7 × 8 ÷ 2 = 56 ÷ 2 = 28
Check. Add the numbers 1 to 7 directly: 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28. ✓
Watch out. Multiply n by (n + 1) first, and only then divide by 2.
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. Use the nth-term rule for powers of 2.
- Remember what powers of 2 are. The powers of 2 start at 2 and double each time: 2n means n twos multiplied together.nth term = 2n
- Replace n by 8. We want the term in position 8, so use 8 in place of every n.8th term = 28
- Work it out. Follow the order of operations: brackets, powers, multiply and divide, then add and subtract.8th term = 28 = 256
Check. The sequence 2, 4, 8, 16, … matches the start of the powers of 2, and the 8th term follows the same rule. ✓
Watch out. Do not mix up 2n with 2 × n.
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. Use the nth-term rule for square numbers.
- Remember what square numbers are. A square number is a whole number multiplied by itself. It is the number of small squares in a bigger square with that side length.nth term = n2
- Replace n by 15. We want the term in position 15, so use 15 in place of every n.15th term = 152
- Work it out. Follow the order of operations: brackets, powers, multiply and divide, then add and subtract.15th term = 152 = 225
Check. The sequence 1, 4, 9, 16, … matches the start of the square numbers, and the 15th term follows the same rule. ✓
Watch out. Do not confuse n² with 2n: 5² = 25 but 2 × 5 = 10.
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. Use the nth-term rule for cube numbers.
- Remember what cube numbers are. A cube number is a whole number multiplied by itself, and then by itself again. It is the number of small cubes in a bigger cube with that side length.nth term = n3
- Replace n by 6. We want the term in position 6, so use 6 in place of every n.6th term = 63
- Work it out. Follow the order of operations: brackets, powers, multiply and divide, then add and subtract.6th term = 63 = 216
Check. The sequence 1, 8, 27, 64, … matches the start of the cube numbers, and the 6th term follows the same rule. ✓
Watch out. Do not confuse n² with 2n: 5² = 25 but 2 × 5 = 10.
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
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 12 = 1
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 22 = 4
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 32 = 9
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 42 = 16
- Replace n by 5 to find the 5th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).5th term = 52 = 25
- Write the terms as a list. 1, 4, 9, 16, 25
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 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
- Replace n by 5 to find the 5th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).5th term = 53 = 125
- Write the terms as a list. 1, 8, 27, 64, 125
Check. Re-do the last one on a calculator: the 5th term should be 125. ✓
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 = n(n + 1)2
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 1 × (1 + 1) ÷ 2 = 1 × 2 ÷ 2 = 2 ÷ 2 = 1
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 2 × (2 + 1) ÷ 2 = 2 × 3 ÷ 2 = 6 ÷ 2 = 3
- 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) ÷ 2 = 3 × 4 ÷ 2 = 12 ÷ 2 = 6
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 4 × (4 + 1) ÷ 2 = 4 × 5 ÷ 2 = 20 ÷ 2 = 10
- Replace n by 5 to find the 5th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).5th term = 5 × (5 + 1) ÷ 2 = 5 × 6 ÷ 2 = 30 ÷ 2 = 15
- Write the terms as a list. 1, 3, 6, 10, 15
Check. Re-do the last one on a calculator: the 5th term should be 15. ✓
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
- Replace n by 1 to find the 1st term. Work out powers first, then multiplications, then add or subtract (BIDMAS).1st term = 21 = 2
- Replace n by 2 to find the 2nd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).2nd term = 22 = 4
- Replace n by 3 to find the 3rd term. Work out powers first, then multiplications, then add or subtract (BIDMAS).3rd term = 23 = 8
- Replace n by 4 to find the 4th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).4th term = 24 = 16
- Replace n by 5 to find the 5th term. Work out powers first, then multiplications, then add or subtract (BIDMAS).5th term = 25 = 32
- Write the terms as a list. 2, 4, 8, 16, 32
Check. Re-do the last one on a calculator: the 5th term should be 32. ✓
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. Recognise the sequence by its differences or ratios and by comparing with the special sequences you know.
- Find the differences between neighbouring terms. This tells us what kind of sequence it is.4 − 1 = 3
9 − 4 = 5
16 − 9 = 7
These are 3, 5, 7: the odd numbers going up by 2 each time. - Compare with the special sequences you know. It helps to know these by heart: squares 1, 4, 9, 16, 25; cubes 1, 8, 27, 64, 125; triangular numbers 1, 3, 6, 10, 15; powers of 2: 2, 4, 8, 16; powers of 10: 10, 100, 1000.The square numbers are 1, 4, 9, 16, 25, 36, 49, …
Our sequence 1, 4, 9, 16 appears in this list. - Use the rule for the square numbers to find the next 2 terms. The rule is nth term = n2.5th term of the square numbers: 52 = 25
6th term of the square numbers: 62 = 36
Check. The next terms 25, 36 carry on the same pattern as 9, 16. ✓
Watch out. A sequence that looks like squares might be something else. Check at least three terms, and use the differences.
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. Recognise the sequence by its differences or ratios and by comparing with the special sequences you know.
- Find the differences between neighbouring terms. This tells us what kind of sequence it is.8 − 1 = 7
27 − 8 = 19
64 − 27 = 37
These are 7, 19, 37: going up by a bigger amount each time (the differences 7, 19, 37, … are not simple). - Compare with the special sequences you know. It helps to know these by heart: squares 1, 4, 9, 16, 25; cubes 1, 8, 27, 64, 125; triangular numbers 1, 3, 6, 10, 15; powers of 2: 2, 4, 8, 16; powers of 10: 10, 100, 1000.The cube numbers are 1, 8, 27, 64, 125, 216, 343, …
Our sequence 1, 8, 27, 64 appears in this list. - Use the rule for the cube numbers to find the next 2 terms. The rule is nth term = n3.5th term of the cube numbers: 53 = 125
6th term of the cube numbers: 63 = 216
Check. The next terms 125, 216 carry on the same pattern as 27, 64. ✓
Watch out. A sequence that looks like squares might be something else. Check at least three terms, and use the differences.
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. Recognise the sequence by its differences or ratios and by comparing with the special sequences you know.
- Find the differences between neighbouring terms. This tells us what kind of sequence it is.6 − 3 = 3
10 − 6 = 4
15 − 10 = 5
These are 3, 4, 5: one more each time (2, 3, 4, …). - Compare with the special sequences you know. It helps to know these by heart: squares 1, 4, 9, 16, 25; cubes 1, 8, 27, 64, 125; triangular numbers 1, 3, 6, 10, 15; powers of 2: 2, 4, 8, 16; powers of 10: 10, 100, 1000.The triangular numbers are 1, 3, 6, 10, 15, 21, 28, …
Our sequence 3, 6, 10, 15 appears in this list, starting from the 2nd one. - Use the rule for the triangular numbers to find the next 2 terms. The rule is nth term = n(n + 1)2.6th term of the triangular numbers: 6 × (6 + 1) ÷ 2 = 6 × 7 ÷ 2 = 42 ÷ 2 = 21
7th term of the triangular numbers: 7 × (7 + 1) ÷ 2 = 7 × 8 ÷ 2 = 56 ÷ 2 = 28
Check. The next terms 21, 28 carry on the same pattern as 10, 15. ✓
Watch out. A sequence that looks like squares might be something else. Check at least three terms, and use the differences.
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. Recognise the sequence by its differences or ratios and by comparing with the special sequences you know.
- Find the differences between neighbouring terms. This tells us what kind of sequence it is.9 − 4 = 5
16 − 9 = 7
25 − 16 = 9
These are 5, 7, 9: the odd numbers going up by 2 each time. - Compare with the special sequences you know. It helps to know these by heart: squares 1, 4, 9, 16, 25; cubes 1, 8, 27, 64, 125; triangular numbers 1, 3, 6, 10, 15; powers of 2: 2, 4, 8, 16; powers of 10: 10, 100, 1000.The square numbers are 1, 4, 9, 16, 25, 36, 49, …
Our sequence 4, 9, 16, 25 appears in this list, starting from the 2nd one. - Use the rule for the square numbers to find the next 2 terms. The rule is nth term = n2.6th term of the square numbers: 62 = 36
7th term of the square numbers: 72 = 49
Check. The next terms 36, 49 carry on the same pattern as 16, 25. ✓
Watch out. A sequence that looks like squares might be something else. Check at least three terms, and use the differences.
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. Recognise the sequence by its differences or ratios and by comparing with the special sequences you know.
- Divide each term by the one before it. This tells us what kind of sequence it is.4 ÷ 2 = 2
8 ÷ 4 = 2
16 ÷ 8 = 2
Every term is multiplied by the same number each time. - Compare with the special sequences you know. It helps to know these by heart: squares 1, 4, 9, 16, 25; cubes 1, 8, 27, 64, 125; triangular numbers 1, 3, 6, 10, 15; powers of 2: 2, 4, 8, 16; powers of 10: 10, 100, 1000.The powers of 2 are 2, 4, 8, 16, 32, 64, 128, …
Our sequence 2, 4, 8, 16 appears in this list. - Use the rule for the powers of 2 to find the next 2 terms. The rule is nth term = 2n.5th term of the powers of 2: 25 = 32
6th term of the powers of 2: 26 = 64
Check. The next terms 32, 64 carry on the same pattern as 8, 16. ✓
Watch out. A sequence that looks like squares might be something else. Check at least three terms, and use the differences.
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. Recognise the sequence by its differences or ratios and by comparing with the special sequences you know.
- Divide each term by the one before it. This tells us what kind of sequence it is.100 ÷ 10 = 10
1000 ÷ 100 = 10
Every term is multiplied by the same number each time. - Compare with the special sequences you know. It helps to know these by heart: squares 1, 4, 9, 16, 25; cubes 1, 8, 27, 64, 125; triangular numbers 1, 3, 6, 10, 15; powers of 2: 2, 4, 8, 16; powers of 10: 10, 100, 1000.The powers of 10 are 10, 100, 1000, 10000, 100000, 1000000, 10000000, …
Our sequence 10, 100, 1000 appears in this list. - Use the rule for the powers of 10 to find the next 2 terms. The rule is nth term = 10n.4th term of the powers of 10: 104 = 10000
5th term of the powers of 10: 105 = 100000
Check. The next terms 10000, 100000 carry on the same pattern as 100, 1000. ✓
Watch out. A sequence that looks like squares might be something else. Check at least three terms, and use the differences.
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. Work backwards from the term to the position.
- Remember the rule for square numbers. A square number is a whole number multiplied by itself. It is the number of small squares in a bigger square with that side length.nth term = n2
- Use the square root. The inverse (undo) of squaring is the square root. We need the number that multiplies by itself to give 169.√169 = 13, because 13 × 13 = 169
- Write the answer. The position is the value of n.169 is the 13th term of the square numbers.
Check. Substitute n = 13: 132 = 169. ✓
Watch out. The answer is the position (n), not the term. Say which one you are giving.
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. Work backwards from the term to the position.
- Remember the rule for cube numbers. A cube number is a whole number multiplied by itself, and then by itself again. It is the number of small cubes in a bigger cube with that side length.nth term = n3
- Use the cube root. The inverse of cubing is the cube root. We need the number that gives 343 when multiplied by itself twice more.7 × 7 × 7 = 343, so the cube root of 343 is 7
- Write the answer. The position is the value of n.343 is the 7th term of the cube numbers.
Check. Substitute n = 7: 73 = 343. ✓
Watch out. The answer is the position (n), not the term. Say which one you are giving.
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. Work backwards from the term to the position.
- Remember the rule for triangular numbers. A triangular number is the number of dots in a triangle with rows of 1, 2, 3, … dots. The nth triangular number is 1 + 2 + … + n. Two copies of the same triangle fit together to make a rectangle n by (n + 1), so one copy is half of that: n(n + 1) ÷ 2.nth term = n(n + 1)2
- Build the triangular numbers until you reach it. Add one more each time: add 1, then 2, then 3, …1st: 1
2nd: 3
3rd: 6
4th: 10
5th: 15
6th: 21
7th: 28
8th: 36
9th: 45
10th: 55 - Write the answer. The position is the value of n.55 is the 10th term of the triangular numbers.
Check. Substitute n = 10: 10 × (10 + 1) ÷ 2 = 10 × 11 ÷ 2 = 110 ÷ 2 = 55. ✓
Watch out. The answer is the position (n), not the term. Say which one you are giving.
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. Work backwards from the term to the position.
- Remember the rule for powers of 2. The powers of 2 start at 2 and double each time: 2n means n twos multiplied together.nth term = 2n
- Keep multiplying until you reach it. Start from 2 and multiply by 2 each time.1st: 2
2nd: 4
3rd: 8
4th: 16
5th: 32
6th: 64 - Write the answer. The position is the value of n.64 is the 6th term of the powers of 2.
Check. Substitute n = 6: 26 = 64. ✓
Watch out. The answer is the position (n), not the term. Say which one you are giving.
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. Work backwards from the term to the position.
- Remember the rule for powers of 10. The powers of 10 start at 10 and multiply by 10 each time: 10n means n tens multiplied together.nth term = 10n
- Keep multiplying until you reach it. Start from 10 and multiply by 10 each time.1st: 10
2nd: 100
3rd: 1000 - Write the answer. The position is the value of n.1000 is the 3rd term of the powers of 10.
Check. Substitute n = 3: 103 = 1000. ✓
Watch out. The answer is the position (n), not the term. Say which one you are giving.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. A Fibonacci-type sequence is built by adding the previous two terms.
- Write the first two terms. A Fibonacci-type sequence needs two starting terms.1, 1
- Add the last two terms to make each new term. In a Fibonacci-type sequence every term is the sum of the two terms before it.Term 3 = 1 + 1 = 2
Term 4 = 1 + 2 = 3
Term 5 = 2 + 3 = 5
Term 6 = 3 + 5 = 8
Term 7 = 5 + 8 = 13
Term 8 = 8 + 13 = 21 - Write the sequence. 1, 1, 2, 3, 5, 8, 13, 21
Check. Check the last term: 8 + 13 = 21. ✓
Watch out. Always add the last two terms, not the first two.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. A Fibonacci-type sequence is built by adding the previous two terms.
- Write the first two terms. A Fibonacci-type sequence needs two starting terms.2, 3
- Add the last two terms to make each new term. In a Fibonacci-type sequence every term is the sum of the two terms before it.Term 3 = 2 + 3 = 5
Term 4 = 3 + 5 = 8
Term 5 = 5 + 8 = 13
Term 6 = 8 + 13 = 21
Term 7 = 13 + 21 = 34
Term 8 = 21 + 34 = 55 - Write the sequence. 2, 3, 5, 8, 13, 21, 34, 55
Check. Check the last term: 21 + 34 = 55. ✓
Watch out. Always add the last two terms, not the first two.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. A Fibonacci-type sequence is built by adding the previous two terms.
- Write the first two terms. A Fibonacci-type sequence needs two starting terms.1, 3
- Add the last two terms to make each new term. In a Fibonacci-type sequence every term is the sum of the two terms before it.Term 3 = 1 + 3 = 4
Term 4 = 3 + 4 = 7
Term 5 = 4 + 7 = 11
Term 6 = 7 + 11 = 18
Term 7 = 11 + 18 = 29
Term 8 = 18 + 29 = 47 - Write the sequence. 1, 3, 4, 7, 11, 18, 29, 47
Check. Check the last term: 18 + 29 = 47. ✓
Watch out. Always add the last two terms, not the first two.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. A Fibonacci-type sequence is built by adding the previous two terms.
- Write the first two terms. A Fibonacci-type sequence needs two starting terms.5, 2
- Add the last two terms to make each new term. In a Fibonacci-type sequence every term is the sum of the two terms before it.Term 3 = 5 + 2 = 7
Term 4 = 2 + 7 = 9
Term 5 = 7 + 9 = 16
Term 6 = 9 + 16 = 25
Term 7 = 16 + 25 = 41
Term 8 = 25 + 41 = 66 - Write the sequence. 5, 2, 7, 9, 16, 25, 41, 66
Check. Check the last term: 25 + 41 = 66. ✓
Watch out. Always add the last two terms, not the first two.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Every term is the sum of the previous two. To find a missing term, add the two before it, or subtract to work backwards.
- Use the rule: each term is the sum of the two terms before it. Look for a gap where you know the two terms before it (add them) or the two terms after it (subtract).Terms: 3, 5, ?, 13, ?
- Find term 3. Term 3 is the sum of the two terms before it.3 + 5 = 8
- Find term 5. Term 5 is the sum of the two terms before it.8 + 13 = 21
- Write the complete sequence. 3, 5, 8, 13, 21
Check. Check every term is the sum of the two before it: 3 + 5 = 8, 5 + 8 = 13, 8 + 13 = 21. ✓
Watch out. When working backwards, subtract: the missing term is the later term minus its neighbour.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Every term is the sum of the previous two. To find a missing term, add the two before it, or subtract to work backwards.
- Use the rule: each term is the sum of the two terms before it. Look for a gap where you know the two terms before it (add them) or the two terms after it (subtract).Terms: ?, ?, 7, 11, 18
- Find term 2. Term 4 is the sum of terms 2 and 3, so term 2 = term 4 − term 3. We are working backwards.11 − 7 = 4
- Find term 1. Term 3 is the sum of terms 1 and 2, so term 1 = term 3 − term 2. We are working backwards.7 − 4 = 3
- Write the complete sequence. 3, 4, 7, 11, 18
Check. Check every term is the sum of the two before it: 3 + 4 = 7, 4 + 7 = 11, 7 + 11 = 18. ✓
Watch out. When working backwards, subtract: the missing term is the later term minus its neighbour.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Every term is the sum of the previous two. To find a missing term, add the two before it, or subtract to work backwards.
- Use the rule: each term is the sum of the two terms before it. Look for a gap where you know the two terms before it (add them) or the two terms after it (subtract).Terms: 1, ?, ?, 7, 11
- Find term 3. Term 5 is the sum of terms 3 and 4, so term 3 = term 5 − term 4. We are working backwards.11 − 7 = 4
- Find term 2. Term 4 is the sum of terms 2 and 3, so term 2 = term 4 − term 3. We are working backwards.7 − 4 = 3
- Write the complete sequence. 1, 3, 4, 7, 11
Check. Check every term is the sum of the two before it: 1 + 3 = 4, 3 + 4 = 7, 4 + 7 = 11. ✓
Watch out. When working backwards, subtract: the missing term is the later term minus its neighbour.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Every term is the sum of the previous two. To find a missing term, add the two before it, or subtract to work backwards.
- Use the rule: each term is the sum of the two terms before it. Look for a gap where you know the two terms before it (add them) or the two terms after it (subtract).Terms: 2, 5, ?, 12, ?
- Find term 3. Term 3 is the sum of the two terms before it.2 + 5 = 7
- Find term 5. Term 5 is the sum of the two terms before it.7 + 12 = 19
- Write the complete sequence. 2, 5, 7, 12, 19
Check. Check every term is the sum of the two before it: 2 + 5 = 7, 5 + 7 = 12, 7 + 12 = 19. ✓
Watch out. When working backwards, subtract: the missing term is the later term minus its neighbour.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Every term is the sum of the previous two. To find a missing term, add the two before it, or subtract to work backwards.
- Use the rule: each term is the sum of the two terms before it. Look for a gap where you know the two terms before it (add them) or the two terms after it (subtract).Terms: ?, ?, ?, 11, 18
- Find term 3. Term 5 is the sum of terms 3 and 4, so term 3 = term 5 − term 4. We are working backwards.18 − 11 = 7
- Find term 2. Term 4 is the sum of terms 2 and 3, so term 2 = term 4 − term 3. We are working backwards.11 − 7 = 4
- Find term 1. Term 3 is the sum of terms 1 and 2, so term 1 = term 3 − term 2. We are working backwards.7 − 4 = 3
- Write the complete sequence. 3, 4, 7, 11, 18
Check. Check every term is the sum of the two before it: 3 + 4 = 7, 4 + 7 = 11, 7 + 11 = 18. ✓
Watch out. When working backwards, subtract: the missing term is the later term minus its neighbour.
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. Use the nth-term rule for triangular numbers.
- Remember what triangular numbers are. A triangular number is the number of dots in a triangle with rows of 1, 2, 3, … dots. The nth triangular number is 1 + 2 + … + n. Two copies of the same triangle fit together to make a rectangle n by (n + 1), so one copy is half of that: n(n + 1) ÷ 2.nth term = n(n + 1)2
- Replace n by 9. We want the term in position 9, so use 9 in place of every n.9th term = 9 × (9 + 1) ÷ 2
- Work it out. Follow the order of operations: brackets, powers, multiply and divide, then add and subtract.9th term = 9 × (9 + 1) ÷ 2 = 9 × 10 ÷ 2 = 90 ÷ 2 = 45
Check. Add the numbers 1 to 9 directly: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45. ✓
Watch out. Multiply n by (n + 1) first, and only then divide by 2.
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. Work backwards from the term to the position.
- Remember the rule for triangular numbers. A triangular number is the number of dots in a triangle with rows of 1, 2, 3, … dots. The nth triangular number is 1 + 2 + … + n. Two copies of the same triangle fit together to make a rectangle n by (n + 1), so one copy is half of that: n(n + 1) ÷ 2.nth term = n(n + 1)2
- Build the triangular numbers until you reach it. Add one more each time: add 1, then 2, then 3, …1st: 1
2nd: 3
3rd: 6
4th: 10
5th: 15
6th: 21
7th: 28
8th: 36
9th: 45
10th: 55
11th: 66 - Write the answer. The position is the value of n.66 is the 11th term of the triangular numbers.
Check. Substitute n = 11: 11 × (11 + 1) ÷ 2 = 11 × 12 ÷ 2 = 132 ÷ 2 = 66. ✓
Watch out. The answer is the position (n), not the term. Say which one you are giving.
Method. Work out the number of small cubes in each large cube, then subtract.
- Cubes in the 5 by 5 by 5 cube. This is the 5th cube number.5 × 5 × 5 = 125
- Cubes in the 6 by 6 by 6 cube. This is the 6th cube number.6 × 6 × 6 = 216
- Subtract. The extra cubes needed are the difference.216 − 125 = 91
Check. 125 + 91 = 216. ✓
Watch out. Do not just work out 6 − 5 = 1. The cubes are three-dimensional, so the number needed is much larger.
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. Use the nth-term rule for powers of 2.
- Remember what powers of 2 are. The powers of 2 start at 2 and double each time: 2n means n twos multiplied together.nth term = 2n
- Replace n by 10. We want the term in position 10, so use 10 in place of every n.10th term = 210
- Work it out. Follow the order of operations: brackets, powers, multiply and divide, then add and subtract.10th term = 210 = 1024
Check. The sequence 2, 4, 8, 16, … matches the start of the powers of 2, and the 10th term follows the same rule. ✓
Watch out. Do not mix up 2n with 2 × n.
Method. Add the two triangular numbers, then see if the answer is a square number.
- Add them. Use the triangular numbers 1, 3, 6, 10, 15, 21, …3 + 6 = 9
- Compare with the square numbers. Square numbers: 1, 4, 9, 16, 25, 36, …9 = 3 × 3 = 3², a square number
Check. Two triangles of dots fit together to make a square, with one extra row of dots. ✓
Watch out. The pattern is that two consecutive triangular numbers always make a square number.
Method. Add the two triangular numbers, then see if the answer is a square number.
- Add them. Use the triangular numbers 1, 3, 6, 10, 15, 21, …6 + 10 = 16
- Compare with the square numbers. Square numbers: 1, 4, 9, 16, 25, 36, …16 = 4 × 4 = 4², a square number
Check. Two triangles of dots fit together to make a square, with one extra row of dots. ✓
Watch out. The pattern is that two consecutive triangular numbers always make a square number.
Method. Add the two triangular numbers, then see if the answer is a square number.
- Add them. Use the triangular numbers 1, 3, 6, 10, 15, 21, …10 + 15 = 25
- Compare with the square numbers. Square numbers: 1, 4, 9, 16, 25, 36, …25 = 5 × 5 = 5², a square number
Check. Two triangles of dots fit together to make a square, with one extra row of dots. ✓
Watch out. The pattern is that two consecutive triangular numbers always make a square number.
Method. List the triangular numbers up to 50 and see if 50 appears.
- Write the triangular numbers. Start with 1 and add 2, then 3, then 4, …1, 3, 6, 10, 15, 21, 28, 36, 45, 55
- Compare. The list jumps from 45 to 55.50 is between 45 and 55, so it is skipped. It is not a triangular number.
Check. n(n + 1) ÷ 2 = 50 would need n(n + 1) = 100, but 9 × 10 = 90 and 10 × 11 = 110. ✓
Watch out. Check numbers either side of the number you are testing.
Method. Find a value of n where the two sequences are different.
- Write the first terms of each. Replace n by 1, 2, 3, 4.2n: 2, 4, 8, 16
n²: 1, 4, 9, 16 - Find a difference. They match at n = 2 and at n = 4, but not at n = 3.When n = 3: 23 = 2 × 2 × 2 = 8 but 3² = 3 × 3 = 9. They are different.
Check. At n = 1 they are also different: 21 = 2 but 1² = 1. ✓
Watch out. One counter-example is enough to show a statement is wrong.