Term-to-term rule $\text{+ or }-\text{ the same each time}$the rule that gets you from one term to the NEXT, e.g. "add 4"
Common difference $\text{next term}-\text{this term}$subtract consecutive terms; if it is constant the sequence is LINEAR (arithmetic)
Square numbers $1,\ 4,\ 9,\ 16,\ 25,\ \ldots$$n\times n$; the gaps go up by $2$ each time
Triangular numbers $1,\ 3,\ 6,\ 10,\ 15,\ \ldots$add $2$, then $3$, then $4$, … (the gaps increase by $1$)
Fibonacci-type $\text{add the previous two terms}$e.g. $2,\ 3,\ 5,\ 8,\ 13,\ \ldots$
Continue · $5,\ 8,\ 11,\ \ldots$
rule: add $3$
next terms $14,\ 17,\ 20$
Missing · $3,\ \square,\ 11,\ 15$
difference is $+4$
missing term $=3+4=7$
Fibonacci · $2,\ 5,\ 7,\ 12,\ \ldots$
add the previous two: $7+12$
next term $=19$
Sequence: a list of numbers following a rule.
Term: a single number in the sequence.
Term-to-term rule: how to get from one term to the next (e.g. add $3$).
Common difference: the constant amount added each time in a linear sequence.
Triangular numbers: $1, 3, 6, 10, \ldots$ – the gaps go up by $1$ each time.
✗ Continuing $2, 4, \ldots$ as $2, 4, 8, 16$ (doubling)
✓ check the RULE first – $2, 4, 6, 8$ (add $2$) is just as likely; use the given terms.
✗ A decreasing sequence has no rule
✓ the difference is just negative, e.g. $20, 17, 14$ is "subtract $3$".
✗ Square numbers $=1, 2, 3, 4$
✓ they are $1, 4, 9, 16$ ($n\times n$).
✗ Fibonacci: adding a constant
✓ add the PREVIOUS TWO terms each time, not a fixed number.
• A sequence with a CONSTANT difference is linear (arithmetic).
• The gaps between square numbers are the odd numbers $3, 5, 7, \ldots$
• In a Fibonacci-type sequence each term is the sum of the previous two.
• Write the differences between terms underneath – a constant difference means "add that each time".
• If the differences are not constant, check the second differences or look for a special sequence.
• Learn the square and triangular numbers by heart.
mathedup.co.uk · sheet M38R