Geometric Progression (GP) is a form of sequence in mathematics in which each of the next term is created by the multiplication of each preceding term by a pre-determined integer known as a common ratio. This progression is also known as a pattern-following geometric sequence of integers.
The common ratio is the ratio between two numbers in a geometric series. It is called a common ratio because it is identical for each number or common, and it is also the ratio between two consecutive numbers in the sequence.
In a geometric progression, the next term of the sequence is produced when we multiply a constant or the common ratio (usually a non-zero integer) to the preceding term. A geometric progression, in general, can be written as:
x, xr, xr2, xr3, xr4, …………. xrn, where x is the first number or item of the geometric progression, and r is the common ratio of the entire GP. We can say that this sequence is in geometric progression because, when we divide the second term from the first, we get xr / x = r. Similarly, if we divide the third term from the second, we get xr2 / xr = r, and if we keep going, we always get the common term r, proving that the above sequence is in geometric progression.
Some more examples of a geometric progression are:
1/y, 1/yr, 1/yr2, 1/yr3, …….1/yrn, here the first term is 1/y, and the common ratio is 1/r. This type of GP is ascending geometric progression.
xrn, ………. xr6, xr4, xr2, x, here xrn is the first term, and x is the last term. This type of GP is a descending geometric progression.
There are two types of geometric progression; they are listed below:
The geometric series with a finite number of terms is known as a finite geometric progression. It is the sequence in which the last phrase is specified. For instance, 1/2,1/4,1/8,1/16, ……….1/1024 is a finite geometric series, with the last term being 1/1024.
The geometric series with an unlimited number of terms is known as infinite geometric progression. It is the sequence in which the final phrase is undefined. For example, 3, 12, 48... is an infinite series with no specified end term.
Suppose we need to find the 29th term of a geometric progression. If we try to calculate the term, it will take a lot of time. Instead, by using the properties of GP, mathematicians have derived a way to find any term inside a GP.
Let x be the first term of the GP, and the defined common ratio is s. We know that the second term will be xs, and the third will be xs2; if we continue this till the nth term, we get xsn-1. Thus, to calculate any term, use the formula xn= xsn-1.
If the GP has a finite number of terms, then the sum is Sn = x(sn–1)/s−1, here ‘s’ is never equal to 1. If the GP has infinite terms, then the sum changes to S∞= x/1-s.