Exploring Geometric Progressions: Formulas, Sums, and Comparisons
Comprehensive Definition, Description, Examples & Rules
Introduction
In mathematics, GP is a type of sequence where the following term is produced by multiplying each preceding term with a fixed number. The fixed number that you multiply is also known as the common ratio. Geometric progression in maths has a great role to play.
Characteristics
The primary characteristics of a geometric sequence formula are:
- Each term of a geometric sequence progression is equal to the previous term if multiplied by the common factor of the sum.
- The multiplier of the common factor is always a non-zero multiplier.
- A geometric sequence can have both finite and infinite terms.
Examples
A geometric progression is generally a sequence of terms, and each succeeding term can directly be generated by multiplying the preceding term with a constant common ratio. The common examples of a geometric progression are:
- 2,4,8,16,32,64
The application of a geometric progression is that it helps to calculate the size of exponential growth, and it especially helps in calculating population growth. Famous mathematicians use the geometric series, which plays a good role in physics and Engineering.
Geometric Sequence Formula
You can use the nth term as a formula for a geometric sequence, and you can use it to find the exact thing you want. The geometric sequence formula that you are to use is:
- Tn = arn-1
Here, ‘a’ is the first term of the GP series. ‘r’ is the common ratio of the common factor used to define the series. ‘n’ is the term you want to find out.
The common ratio is the most crucial thing in the geometric progression as it is the ratio between any term in the sequence divided by its previous term. The common ratio is the number that differentiates all the numbers in a geometric progression system. The common ratio of the progression can be both finite and infinite but cannot be zero at any point in time.
You can find any number of the geometric series using the formula mentioned above. You have to know which number you are finding, and then knowing the first number of the series can also be very beneficial. The most important thing for you to find out numbers in a geometric series is to know the exact common ratio of the series.
The sum of Geometric Progression
The sum of geometric progression is the total of all the numbers in a sequence. The sum is very important for finding as it is the exact value you must determine from the sequence. These may be the sum of a few numbers or terms of a geometric progression or all the terms of that progression.
The geometric progression formula for finite geometric series is:
- Sn = a(1 – rn) / (1 – r)
You have to follow the equation to get the sum of any number of the series, but in this case, the common ratio cannot be equal to zero and has to be less than one.
- Sn = a(rn – 1)
This formula will be used if the common ratio is more than one.
When the sequence of the geometric progression approaches infinity, then the behavior of The infinite series will have an infinite sum. In that case, the appropriate notation you will use will be infinity. The sum of any series can only be found that consists of a particular number of terms.
Geometric vs Arithmetic Progression
A primary difference between an arithmetic sequence and a Geometric sequence is that an AP is a set of numbers in which each new phrase differs from its previous term with a fixed amount. A geometric sequence is a new element obtained by multiplying the preceding number with a constant common ratio.
The basic similarities between AP and GP are that they both follow a specific pattern and cannot be broken at any point in time. You can find the sum of both these terms effectively whenever required.
Real-life examples:
- Geometric Progression: If each man decides not to have more children in the present population, the total annual population increase becomes geometric.
- Arithmetic Progression: Saving or investing money in equal installments.
Meaning of Arithmetic Sequence
An Arithmetic sequence meaning is a sequence where all the term numbers are obtained by adding a particular fixed number to the previous term to determine the result of the new term. The primary characteristic of an arithmetic sequence is to find out if a particular set of numbers is a group of percentages. You can also use it to find annual income of some kind.
The common difference is an essential part of the Arithmetic sequence as you are to use it to find the exact sequence of the terms. A common difference is a major difference between two consecutive terms of the sequence adding, which will give you the new sequence.
Defining an arithmetic vs geometric progression sequence is very easy, as an arithmetic sequence is when you use a fixed number to find out the new numbers of the sentence. Geometric progression is a sequence with a common ratio you must follow to find out the new numbers.
Derivation of the Sum of Geometric Series
The overall sum of geometric series derivation can be found with the formula
- Sn = a(rn – 1)
‘a’ is the first term of the series, ‘r’ is the direct common ratio, and ‘n’ is the number of terms.
The mathematical sequence involves the geometric progression, and a fixed number, the common ratio, precedes it. One of the primary principles of mathematics is to use geometric progressions to determine a series of numbers that can do a particular thing. The geometric progression will help you find information in a large amount, such as the annual population and things like this.
The examples of a sum of a geometric progression are:
Summing out the 2,4,8,16,32 then,
- Sn = 2( 2⁵ – 1) i.e. 62
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Key Takeaways
- Arithmetic and geographic progressions greatly impact real life and are not only numbers but are problem-solving techniques.
- The common difference and ratio are the two most important aspects of the progression sequence, which helps create the exact sequence.
- Understanding the geometric progression and the common ratio is important for advanced mathematics.
- To determine the sum of any sequence, you need to find the total number of terms in that particular sequence.
Quiz
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Frequently Asked Questions
To find out the common ratio in a geometric progression, you need to divide the nth term of the sequence by (n – 1)th term. You will find the common ratio, the ratio by which all the sequence numbers are divided.
The specific formula that you need to use to find out the nth of an arithmetic sequence is:
- Nth term = a +( n – 1)d
Here ‘a’ is the first term of the sequence, ‘n’ is the term you want to find out, and ‘d’ is the common difference.
The formula that you are supposed to use for finding out the nth term of a geometric sequence is:
- Tn = arn-1
Here, ‘a’ is the first term of the GP series. ‘r’ is the common ratio of the common factor used to define the series. ‘n’ is the term you want to find out.
To find out the sum of an arithmetic sequence, you need a total number of terms to find out its sum. The formula that you are supposed to use for the sequence is:
- Sn = n/2 [ 2a + (n – 1)d]
Or
- Sn = n/2 [a1 + an)
For finding out the sum of a geometric progression, the formula that you are supposed to use are:
Sn = a(1 – rn) / (1 – r)
You have to follow the equation to get the sum of any number of the series, but in this case, the common ratio cannot be equal to zero and has to be less than one.
Sn = a(rn – 1)
This formula will be used if the common ratio is more than one. You can find the exact answer for the geometric progression sum using the formulas.
The real-life applications that you want to follow for both of these sequences are:
- Geometric: The disintegration of each radioactive component happens in a geometric sequence.
- Arithmetic: Traveling in a car at a constant speed.
