Geometric Progression (GP)

A geometric progression (GP) is a type of series that is characterized by a common ratio and is of the form a, ar, ar^2, ar^3, ... where 'a' is the first term and 'r' is the common ratio.

Explanation:

1. Geometric Progression:
- A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
- In a GP, the ratio of any term to its preceding term is always constant.
- The general form of a GP is given by a, ar, ar^2, ar^3, ... where 'a' is the first term and 'r' is the common ratio.
- The common ratio 'r' can be any non-zero real number.

2. Arithmetic Progression:
- An arithmetic progression (AP) is a sequence of numbers where each term after the first is obtained by adding a constant difference to the preceding term.
- In an AP, the difference between any term and its preceding term is always constant.
- The general form of an AP is given by a, a+d, a+2d, a+3d, ... where 'a' is the first term and 'd' is the common difference.
- The common difference 'd' can be any real number.

Comparison:

- In an arithmetic progression (AP), the terms are obtained by adding a constant difference, whereas in a geometric progression (GP), the terms are obtained by multiplying a constant ratio.
- The terms in an AP have a linear relationship, while the terms in a GP have an exponential relationship.
- The common difference in an AP can be any real number, whereas the common ratio in a GP can be any non-zero real number.

Conclusion:

The series described in the given form, a, ar, ar^2, ar^3, ..., is a geometric progression (GP) as it follows the pattern of multiplying each term by a constant ratio 'r'. Therefore, the correct answer is option 'A' - Geometric Progression (GP).