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Each term in a geometric sequence or progression is equal to the previous term multiplied by the common factor, which is a constant non-zero multiplier. Geometric sequences can have a finite number of terms or an infinite number of terms. The terms of a geometric sequence can quickly become very large, very negative, or very close to zero in either case. The terms change much more quickly in geometric sequences than in arithmetic sequences, but unlike infinite arithmetic sequences, geometric sequences can approach zero depending on the common factor.
Geometric progression
A geometric sequence or progression abbreviated as G.P is a set of numbers in which each subsequent term is obtained by multiplying the previous term by a fixed number. The common ratio is a fixed number that is usually denoted by r.
There are two types of geometric progression.
1. Finite geometric progression
There are a finite number of terms in a finite geometric progression. It is at this point in the progression that the last term is defined.
For example, a finite geometric series with the last term 1/32768 is 1/2,1/4,1/8,1/16,…,1/32768.
2. Infinite Geometric Progression
The opposite of a finite geometric progression series is infinite geometric progression series. It’s a series with an infinite number of terms, which means the series’ final term isn’t defined. The general expression can be used to describe it.
For example, 3, 6, +12, 24, +… is an infinite series with no defined last term.
Formula of Geometric Progression
The nth term in the sequence is found using the geometric progression formula. A common ratio, or ‘r,’ and the first term, ‘a,’ are required. This is the formula:
an = arn-1
Geometric Progression properties
Geometric Progression’s properties are given below:
In terms of the geometric mean, geometric sequences have unique properties. The square root of two numbers’ product is their geometric mean. Because the product 5×20 = 100 and the square root of 100 is 10, the geometric mean of 5 and 20 is 10.
Each term in a geometric sequence is the geometric mean of the terms before it and after it. For example, 6 is the geometric mean of 3 and 12, 12 is the geometric mean of 6 and 24, and 24 is the geometric mean of 12 and 48 in the sequence 3, 6, 12…
Other geometric sequence properties are influenced by the common factor. Infinite geometric sequences will approach positive infinity if the common factor r is greater than 1. The sequences will approach zero if r is between 0 and 1. The sequences will approach zero if r is between 0 and 1, but the terms will alternate between positive and negative values. If r is less than 1, the terms will alternate between positive and negative values, trending toward both positive and negative infinity.
Geometric sequences and their properties are particularly useful in mathematical and scientific models of real-world processes. The study of populations that grow at a constant rate over time or investments that earn interest can benefit from the use of specific sequences. Based on the starting point and the common factor, the general and recursive formulas allow for accurate future predictions.
Geometric Progression example
Which of the following G. P. terms is 6, –12, 24, – 48, … is 384?
Solution: Let nth be the term
(ar)n-1 = 384
a = 6, and r = (-2)
Calculating n:
6(-2)n-1 = 384
(-2)n-1 = 64
n-1 = 6
n = 7
Conclusion
We conclude in this article that in a geometric progression, each subsequent term is obtained by multiplying the common ratio by the term before it. Mathematicians use geometric series all the time. When each term of a Geometric Progression is multiplied or divided by the same non-zero quantity, the new series has the same common ratio as the original. Physics, engineering, biology, economics, computer science, queuing theory, and finance all benefit from them. Geometric series are one of the simplest examples of infinite series with finite sums, though this property does not apply to all of them.
