The phrasing “which of the following is an arithmetic sequence apex” refers to selecting the option whose consecutive terms change by the same constant amount. That constant change is the common difference \(d\).
Arithmetic sequence criterion
A sequence \(\{a_n\}\) is arithmetic when the difference between consecutive terms is constant:
When a constant \(d\) exists, the explicit (nth-term) form is linear in \(n\):
Multiple-choice options
The options below illustrate a typical A–D format.
| Option | Sequence (first four terms) |
|---|---|
| A | 2, 5, 8, 11, … |
| B | 1, 2, 4, 8, … |
| C | 0, 1, 4, 9, … |
| D | 10, 7, 3, −2, … |
Consecutive-difference check
The common-difference test compares successive subtractions within each option.
| Option | Differences | Arithmetic status |
|---|---|---|
| A | \(5-2=3,\; 8-5=3,\; 11-8=3\) | Constant \(d=3\) |
| B | \(2-1=1,\; 4-2=2,\; 8-4=4\) | Not constant |
| C | \(1-0=1,\; 4-1=3,\; 9-4=5\) | Not constant |
| D | \(7-10=-3,\; 3-7=-4,\; -2-3=-5\) | Not constant |
Only option A maintains one repeated difference across all shown gaps, so it is the arithmetic sequence.
Visualization of linear pattern
Common pitfalls
- Constant ratio confusion: geometric sequences keep a constant ratio \( \frac{a_{n+1}}{a_n} \), while arithmetic sequences keep a constant difference \(a_{n+1} - a_n\).
- Local agreement trap: two consecutive differences matching once does not guarantee an arithmetic sequence; the same difference must persist across all consecutive pairs.
- Sign handling: decreasing arithmetic sequences have \(d < 0\); constant negativity does not imply arithmetic unless the difference remains constant.
Direct conclusion
Option A is the arithmetic sequence because every consecutive subtraction yields the same common difference \(d=3\).