The key difference between a finite and an infinite arithmetic progression (AP) lies in whether the sequence has a last term or continues indefinitely.
Finite Arithmetic Progression:
- Definition: A finite AP is an arithmetic sequence that has a limited number of terms, meaning it has a definite starting and ending point.
- Last Term: It possesses a last term (also known as the nth term).
- Example: 2, 4, 6, 8, 10. Here, the last term is 10, and the number of terms is 5.
Infinite Arithmetic Progression:
- Definition: An infinite AP is an arithmetic sequence that continues without end. The sequence goes on forever.
- Last Term: It does not have a last term.
- Example: 1, 3, 5, 7, 9, ... The ellipsis (...) indicates that the sequence continues indefinitely.
Table Summarizing the Differences:
| Feature | Finite Arithmetic Progression | Infinite Arithmetic Progression |
|---|---|---|
| Number of Terms | Limited | Unlimited |
| Last Term | Exists | Does not exist |
| Termination | Terminates | Never terminates |
| Practical Usage | More common in practical problems involving specific amounts. | Used in theoretical analysis and some abstract mathematical models. |
In simple terms: Think of a finite AP as a line segment with a clear beginning and end, while an infinite AP is like a ray that starts at a point and extends endlessly in one direction.
