Geometric Progression

A geometric progression (GP), also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In a geometric progression, the ratio of any term and its previous term is equal to a fixed constant, known as the common ratio. The general form of a geometric progression is given by (a, ar, ar^2, ar^3, ar^4, \ldots, ar^{n-1}), where (a) is the first term and (r) is the common ratio.

Properties and Formulas

• The nth term of a geometric sequence with initial value (a = a_1) and common ratio (r) is given by (a_n = ar^{n-1}) .
• The sum of the first n terms of a geometric progression is given by (S_n = a\frac{{1-r^n}}{{1-r}}) when (r \neq 1) and (S_n = na) when (r = 1) .
• The product of a geometric progression is the product of all its terms.

Examples

• An example of a geometric progression is (2, 4, 8, 16, \ldots), where each term is obtained by multiplying the preceding element by a common ratio.
• Another example is the sequence (1, 3, 9, 27, 81), where the next term is obtained by multiplying the preceding element by 3.

In summary, a geometric progression is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. It has various properties and formulas that can be used to calculate specific terms and the sum of the terms in the progression.