C2.7; E2.7 - Sequences and term rules

Continuing patterns

A sequence is an ordered list of numbers called terms. The position matters: in $4,7,10,13,\ldots$, the first term is 4 and the third term is 10. The three dots mean that the pattern continues.

Look for structure before guessing a next term. A picture can explain what is added; in a number list, differences or repeated operations can suggest the intended pattern. Check a proposed rule against every supplied transition. A finite list can fit more than one rule, so use a stated construction or the simplest consistent pattern in the question.

[DIAGRAM: asset_name: Adjoining-square stick pattern; asset_slug: adjoining_square_sticks; caption: The first square uses four sticks. Each adjoining square shares one side and adds three sticks.; description: Four separate rows contain one, two, three and four equal adjoining squares, labelled pattern one to four with stick totals four, seven, ten and thirteen. The first square has four blue edges. In patterns two to four, the newest square's three unshared edges are sage and the earlier sticks are blue. Shared edges are counted once.]

The next example gives the construction instead of only its totals. Explain the extra sticks before calculating the next total.

The totals depend on the construction. Separate squares would use four new sticks each time, while adjoining squares share sides. A diagram's sharing rule is mathematical information.

Use the changes to continue this new pattern.

Term-to-term rules

A term-to-term rule tells you how to obtain a term from the preceding term. “Add 3” describes the stick totals, but the starting term 4 is also needed to identify that sequence. The same rule starting at 1 gives a different sequence.

Test addition by subtracting consecutive terms. Test multiplication by dividing consecutive terms when the earlier term is non-zero. Sometimes the amount added changes: $2,5,9,14,\ldots$ adds $3,4,5,\ldots$. State that changing rule precisely rather than saying only “add”.