Numbers and sequences
Sequences
A sequence is a list of numbers that follow a rule. In Stage 6 you count on and back in fractions and decimals, and use a position-to-term rule to find any term you like.
The term-to-term rule
A term-to-term rule tells you how to get from one term to the next. If you know the first term and the rule, you can write out the sequence. Example: first term 4, rule 'add 7' gives 4, 11, 18, 25, 32, ... If the terms go down, the rule subtracts, like 71, 65, 59, 53 (subtract 6).
The position-to-term rule
The position-to-term rule connects the position number (1st, 2nd, 3rd, ...) straight to the term. It lets you jump to any term without writing them all out. For 4, 8, 12, 16, ... position 1 gives 4, position 2 gives 8, so the rule is 'multiply by 4'. The 25th term is 25 x 4 = 100.
Counting on and back in decimals and fractions
Sequences do not have to use whole numbers. You can count on or back in steps such as 0.4, 0.01 or one quarter. Example: 15, 14.6, 14.2, 13.8, ... counts back in steps of 0.4. Look carefully at the step size before you fill in gaps.
Worked examples
Continue a sequence with a term-to-term rule
- The sequence is 4, 11, 18, 25 and the rule is 'add 7'.
- To find the next term, add 7 to the last term: 25 + 7.
Answer: The next term is 32.
Use a position-to-term rule to find any term
- The terms 4, 8, 12, 16 match positions 1, 2, 3, 4, so the rule is 'multiply by 4'.
- For the 25th term, multiply the position by 4: 25 x 4.
Answer: The 25th term is 100.
Count on in decimals
- Start at 1 and count on in steps of 0.4: 1, 1.4, 1.8, ...
- Keep adding 0.4: 1.8 + 0.4 gives the next term.
Answer: The next term is 2.2.
Count on in fractions
- Pablo counts on in quarters: 1 1/4, 1 1/2, 1 3/4, ...
- Add one quarter to 1 3/4.
Answer: The next term is 2.
Where you see this in real life
- Reading the marks on a measuring jug that count up in steps of 0.1 litres.
- Saving the same amount of pocket money each week and tracking the running total.
- Counting down a timer that ticks back in equal steps.
Key words
- Sequence
- A list of numbers that follow a rule, one after another.
- Term
- One of the numbers in a sequence.
- Term-to-term rule
- The rule that gets you from one term to the next.
- Position-to-term rule
- The rule that turns a position number straight into its term.
- Step
- The amount you add or subtract between terms.