Sequences and functions
Generating sequences (1)
Find what gets you from one term to the next, and you can carry the sequence on as far as you like.
The term-to-term rule
A SEQUENCE is a list of numbers following a pattern. Each number is a TERM. The TERM-TO-TERM RULE tells you how to get from one term to the next. 7, 11, 15, 19, … 7 + 4 = 11, 11 + 4 = 15, 15 + 4 = 19 The rule is 'add 4', so the next two terms are 23 and 27. 40, 35, 30, 25, … 40 − 5 = 35, 35 − 5 = 30 The rule is 'subtract 5', so the next two terms are 20 and 15.
Rules that multiply or divide
The rule does not have to be adding or subtracting. First term 6, rule 'multiply by 2': 6, 12, 24, 48, … First term 60, rule 'divide by 2': 60, 30, 15, 7.5, … To generate a sequence you need TWO things: the first term, and the rule.
Two-step rules
A rule can have two steps. First term 4, rule 'multiply by 2 then add 1': 4 × 2 + 1 = 9 9 × 2 + 1 = 19 19 × 2 + 1 = 39 So the sequence starts 4, 9, 19, 39, … Do the steps in the order given — swapping them gives a different sequence.
Finite and infinite
An INFINITE sequence continues forever. It is written with three dots at the end: 4, 6, 8, 10, … A FINITE sequence stops. It has no dots: 3, 5, 7, 9, 11, 13, 15 The dots are the only clue, so read them carefully.
Worked examples
Continue 7, 11, 15, 19, …
- 7 + 4 = 11 and 11 + 4 = 15, so the rule is 'add 4'.
- 19 + 4 = 23 and 23 + 4 = 27.
Answer: 23 and 27
Continue 40, 35, 30, 25, …
- Each term is 5 less, so the rule is 'subtract 5'.
- 25 − 5 = 20 and 20 − 5 = 15.
Answer: 20 and 15
First term 6, rule 'multiply by 2'
- 6 × 2 = 12, 12 × 2 = 24, 24 × 2 = 48.
Answer: 6, 12, 24, 48, …
First term 60, rule 'divide by 2'
- 60 ÷ 2 = 30, 30 ÷ 2 = 15, 15 ÷ 2 = 7.5.
Answer: 60, 30, 15, 7.5, …
First term 4, rule 'multiply by 2 then add 1'
- 4 × 2 + 1 = 9.
- 9 × 2 + 1 = 19.
- 19 × 2 + 1 = 39.
Answer: 4, 9, 19, 39, …
Is 3, 5, 7, 9, 11, 13, 15 finite or infinite?
- There are no dots at the end.
- So it stops.
Answer: Finite
Where you see this in real life
- Saving £5 a week gives the sequence 5, 10, 15, 20 … — a term-to-term rule of 'add 5'.
- A bacterial culture doubling every hour follows 'multiply by 2', which grows far faster than adding ever would.
Key words
- Sequence
- a list of numbers following a pattern
- Term
- one number in a sequence
- Term-to-term rule
- how to get from one term to the next
- Finite sequence
- a sequence that stops
- Infinite sequence
- a sequence that continues forever
- First term
- the number a sequence starts with
- Rule
- the instruction that generates the next term