Three structural labels
An AP is described by its first term a, signed common difference d, and whether its list stops or continues indefinitely.
Learning goals
- Find signed common difference consistently.
- Identify a, d and finite or infinite structure.
- Find missing consecutive AP terms.
Concept lesson
Later minus earlier
Always compute d=a_(n+1)-a_n. For -7,-9, use -9-(-7)=-2.
Positive, negative and zero d
Positive d gives an increasing AP, negative d a decreasing AP and d=0 a constant AP. All three are valid.
Finite and infinite APs
A finite AP has a last term; an infinite AP continues without a last term. An ellipsis alone must be interpreted from context.
Find a negative common difference
For -1,-3,-5,..., differences are -3-(-1)=-2 and -5-(-3)=-2. Hence d=-2.
Insert two missing terms
In 4,x1,x2,22,28, the final difference is 6, so d=6. Working from 4 gives x1=10 and x2=16; all gaps are 6.
Classify by d sign
For three APs, calculate d using later-minus-earlier and classify each as increasing, decreasing or constant and finite or infinite.
Recover d from separated terms
The seventh term is 12 more than the fourth. Three equal jumps separate them, so 3d=12 and d=4.
One-mark d patterns
For listed terms use later minus earlier. For separated terms, count the number of equal jumps between their positions.
Exit ticket
For 20,15,10,... state a and d and classify it. Then complete 3,x,y,15 as an AP and verify every gap.