Geometric Progression

In mathematics, the progression or sequence is used throughout to solve different problems. This article is focusing on geometric sequence or geometric progression. Evidence indicates evolution of geometric progression is related to ancient Mesopotamia. Basically, this sequence is considered as a set of values in which each new term is calculated by multiplying the preceding term with a common or constant ratio. Apart from mathematics, Geometric progression has its valuable application in physics, engineering, computer science, biology, finance and economics and queuing theory.

Definition of Geometric progression

A sequence or progression in which each term is produced by multiplying a common number or common ratio and also a common ratio can be identified as differentiating between two consecutive terms is called Geometric progression. In this regard, it is observed that the next term of sequence is found or written by multiplying previous term by the common ratio or number of the sequence.

A geometric progression or sequence can be represented by:

p, pr, pr², pr³, pr⁴, pr⁵, pr⁶, pr⁷ and can be produced to infinite terms.

In this progression,

P is the first term and r is said to be a common term or ratio.

Valuable features and properties of geometric Progression

Some of the important features or properties of geometric progression can be represented:

• Three or more non-zero terms are said to be in geometric progression if the sum of first and third terms is equal to a square of second term. It can be represented in formula as:If a, b, c are a sequence then a x c = b²

• Three consecutive terms of a geometric sequence can be formulated as a/r, a, ar; where a is the middle term of three consecutive sequences and r is the common ratio.
• A geometric progression of four consecutive terms can be represented as p/r³, p/r, pr, pr³ where p is the mean term and r is the common difference
• It is found that the product of the term equidistant from the first term and last term is equal every time. It can be justified by a geometric sequence in the form of a, b, c, d, e, f, g, h; the ah = bg = cf = de.
• If it can be written in general form then it can be represented as 1st term × nth term = 2nd term × n-1st term= 3rd term × n-2nd term and so on.

• In this study, it is found that each term of a geometric progression differs from a common ratio.
• If two geometric progressions are multiplied with each other then a new sequence is produced that also lies in geometric progression.
• If each term of a geometric progression is calculated to the power of a non-zero number then the resulting sequence obtained is also found to be following geometric progression.

General form of geometric progression

There must be a general for each series. I(n this regard geometric progression also follows a general form. It can be represented as

g, gr, gpr², gr³,gr⁴,gr⁵, gr⁵, . . . . . . gn;

From the above represented general form of geometric progression, it can be stated as g is first term of geometric progression, r is the common ratio, and gn is the last term of that geometric progression.

How to find a geometric progression

If the first term in the form of a non-zero number is given and a common difference or ratio is also given then geometric progression is easily formed. An example is used here to show how to find geometric progression. In this regard, first of all, the first term and common ratio are assumed.

Let assume first term be “g” and common difference be “r”;

Then, first term; a = g,

Second term; a2 = g x r = gr,

Third term; a3 = gr x r = gr²,

Fourth term; a4 = gr2 x r = gr³,

Similarly, Nth term of a geometric progression; an = grn-1

Therefore, geometric progression can be found by using above method or following the sequence; g, gr, gr², gr³, gr⁴, . . . . gr^n-1

Conclusion

From the above study, it is concluded that geometric progression is a type of sequence in which its term is calculated by multiplying previous terms with a common difference. A progression is found to be a set of numbers that follow a specific set of rules. In this article, geometric progression, its meaning, and its properties are discussed. This article also contains formulas that aim at how to find geometric progression. Apart from these, some applications of geometric progression are also mentioned in this article.