What is the difference between Arithmetic and Geometric progression?

Arithmetic progression (AP) and geometric progression (GP) are two fundamental types of sequences in mathematics, each with its own distinct characteristics:

Arithmetic Progression (AP):

In an arithmetic progression, each term in the sequence is obtained by adding a fixed number called the "common difference" (d) to the previous term.
The general form of an arithmetic progression is: a, a + d, a + 2d, a + 3d, ...
The common difference (d) can be positive, negative, or zero.
Examples of arithmetic progressions include 2, 4, 6, 8, 10 (where d = 2), and -3, -1, 1, 3, 5 (where d = 2).
Geometric Progression (GP):

In a geometric progression, each term in the sequence is obtained by multiplying the previous term by a fixed number called the "common ratio" (r).
The general form of a geometric progression is: a, ar, ar^2, ar^3, ...
The common ratio (r) must be nonzero.
Examples of geometric progressions include 2, 6, 18, 54, 162 (where r = 3), and 5, 10, 20, 40, 80 (where r = 2).
Key differences between AP and GP:

Difference vs. Ratio:

AP uses a constant "common difference" (d) for each term, while GP uses a constant "common ratio" (r) for each term.
Addition vs. Multiplication:

In AP, you add the common difference to each term to generate the next term.
In GP, you multiply the common ratio by each term to generate the next term.
Growth Rate:

AP exhibits linear growth because the difference between consecutive terms is constant.
GP exhibits exponential growth because the ratio between consecutive terms is constant.
Characteristics:

In AP, consecutive terms have a fixed numerical gap between them.
In GP, consecutive terms have a fixed ratio between them.
Both arithmetic and geometric progressions are used in various mathematical and real-world applications. They have specific formulas for finding the nth term and the sum of the first n terms, which can be helpful for solving problems involving these sequences.