1. The language of sequences
Terms and positions
- A sequence is a list of numbers written in a fixed order and governed by a rule
- Each number in the list is a term, and where it sits in the list is its position
- The letter n stands for the position, so n = 1 marks the first term, n = 2 the second, and so on
- Writing the position above each term before you start is the single most useful habit in this topic
- A sequence can also be described by a growing pattern of shapes, in which case counting the dots, sticks or squares in each picture produces the numbers
Two ways of describing a rule
- A term-to-term rule says what to do to one term to reach the next, such as add 9 each time or halve each time
- It is quick to use but only moves one step at a time, so reaching the 60th term means writing out 59 steps
- A position-to-term rule, usually called the nth term, is an expression in n that produces any term directly
- With nth term 6n − 1, substituting n = 1, 2, 3 gives 5, 11, 17, and substituting n = 100 gives 599 in one line
- With nth term n² − 2, the first three terms are −1, 2 and 7, and the 100th term is 9998
- A question asking for a specific far-off term is really asking you to find the nth term first
Worked example
Position, term and the two kinds of rule
A sequence begins 5, 8, 11, 14. Write down the term-to-term rule, find the nth term, and work out the 50th term.
Solution:
- Write the position numbers above the terms: 1, 2, 3, 4
- The gaps are 8 − 5 = 3, 11 − 8 = 3 and 14 − 11 = 3, so the gap is constant
- The term-to-term rule is "add 3", which needs the previous term to use it
- A constant gap of 3 means the nth term contains 3n
- 3n alone gives 3, 6, 9, 12, which is 2 short of every term
- The nth term is 3n + 2, which is a position-to-term rule
- The 50th term is 3 × 50 + 2 = 152, found without listing the terms in between
- Check on a term other than the first: n = 3 gives 11, which matches
2. Continuing a sequence
Look at the gaps first
- Write the differences between consecutive terms underneath the sequence
- Equal differences mean the sequence goes up or down in fixed steps, so add the same amount again
- Differences that themselves form a pattern still tell you what comes next
- In 5, 6, 8, 12, 20 the differences are 1, 2, 4 and 8, doubling at every step; one more doubling gives 16, and 20 + 16 = 36
- Where the differences give nothing, test whether each term is being multiplied by the same number instead
Worked example
Continuing a falling sequence below zero
A sequence begins 30, 22, 14, 6. Write down the term-to-term rule, then find the next two terms.
Solution:
- Write the differences underneath: 22 − 30 = −8, 14 − 22 = −8 and 6 − 14 = −8
- The difference is the same every time, so the sequence goes down in equal steps of 8
- The term-to-term rule is subtract 8
- Keep subtracting 8, carrying on below zero: 6 − 8 = −2
- Subtract 8 again: −2 − 8 = −10
- The next two terms are −2 and −10
- Check: −2 is 8 less than 6, and −10 is 8 less than −2, the same step as before
Number sequences worth recognising
- Square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
- Cube numbers: 1, 8, 27, 64, 125, 216
- Triangular numbers: 1, 3, 6, 10, 15, 21, formed by adding 1, then 2, then 3, and so on
- Prime numbers: 2, 3, 5, 7, 11, 13, which follow no difference pattern at all
- Recognising these on sight turns many sequence questions into a one-line comparison
Worked example
Recognising a familiar sequence
Write down the next two terms of each sequence: (a) 1, 4, 9, 16 (b) 2, 4, 8, 16 (c) 1, 3, 6, 10
Solution:
- (a) The gaps are 3, 5 and 7, which are not constant, so this is not linear
- These are the square numbers 1², 2², 3², 4²
- The next two are 5² and 6², giving 25 and 36
- (b) Each term is double the one before, so the rule is multiplicative rather than additive
- Doubling 16 gives 32, and doubling again gives 64, so the next two are 32 and 64
- (c) The gaps are 2, 3, 4, increasing by one each time
- These are the triangular numbers, and the next gaps are 5 and 6
- The next two terms are 15 and 21
- Check the gaps first every time: a constant gap means adding the same amount, and a constant multiplier means multiplying by the same number
3. Linear sequences
Finding the nth term
- A linear sequence changes by the same amount at every step, and that fixed step is what the nth term is built on
- The nth term takes the form (step)n + (term before the first)
- Find the step from the differences, then work backwards one place from the first term to get the number that follows n
- A sequence that decreases has a negative step, and the number that follows n is then larger than the first term
- Always test the formula on the second or third term before writing it down
Exam tip
Each part of the nth term scores
A two-mark question asking for the nth term of a linear sequence was set in 7 of the 22 papers. One part right, the multiple of n or the number after it, earns one mark: for 7, 12, 17, 22, both 5n + 7 and 3n + 2 score one. So give a full answer even if you doubt one part, then leave 5n + 2 as it is: tidying it into 7n loses a mark.
Worked example
An increasing and a decreasing sequence
Find the nth term of 7, 11, 15, 19, … and of 20, 14, 8, 2, …
Solution:
- For 7, 11, 15, 19 the differences are all 4, so the formula starts 4n
- Going back one step from 7 gives 3, so the nth term is 4n + 3
- Check with n = 3: 4 × 3 + 3 = 15, which matches the third term
- For 20, 14, 8, 2 the differences are all −6, so the formula starts −6n
- Going back one step from 20 gives 26, so the nth term is 26 − 6n
- Check with n = 4: 26 − 24 = 2, which matches the fourth term
Testing whether a number belongs
- Set the nth term equal to the value in question and solve the equation for n
- A whole-number answer means the value is in the sequence, and n tells you which term it is
- A fraction or a decimal answer means the value never appears in the list, and saying so is what earns the mark
- For the sequence with nth term 4n + 3, solving 4n + 3 = 95 gives n = 23, so 95 is the 23rd term
- Solving 4n + 3 = 100 gives n = 24.25, which is not a position, so 100 never appears
4. Quadratic and cubic sequences
Comparing with the squares and the cubes
- A quadratic sequence has n² in its nth term, and its first differences change while the second differences stay equal
- A cubic sequence has n³ in its nth term, and it is the third differences that settle down to a constant
- The fastest route into a straightforward one is to write the square or cube numbers underneath and compare term by term
- A constant gap means adding or subtracting that number, and a constant ratio means multiplying
Worked example
Comparing with a familiar sequence
Find the nth term of 4, 7, 12, 19, 28 and of 3, 10, 29, 66, 127.
Solution:
- Write the square numbers under the first sequence: 1, 4, 9, 16, 25
- Every term is 3 more than the square below it, so the nth term is n² + 3
- Check with n = 5: 25 + 3 = 28, which matches
- Write the cube numbers under the second sequence: 1, 8, 27, 64, 125
- Every term is 2 more than the cube below it, so the nth term is n³ + 2
- Check with n = 4: 64 + 2 = 66, which matches
- A sequence can also be the squares or cubes starting from a later point, in which case the bracket shifts
- 9, 16, 25, 36 are the square numbers from 3² onwards, so the nth term is (n + 2)²