Geometric progression

While preparing for the IIT-JEE, the mathematics topic of geometric progression is crucial. The whole mathematics has the involvement of the properties and formula regarding this topic. This is the reason that students preparing for the exams and the national and state-level selection need to get through this vital portion of the subject. The example sums and the use of formulae in different sums with variables and the constant values highlight these notes among the students. These notes provide detailed FAQs and practice sums, which gives the students more scope to practice.

Definition:

Mathematics is all about series, patterns, and calculations. The geometric progression is the type of series where every term contains a positive and constant common ratio by the preceding term. The terms follow this progression by multiplying each term by a common number. If we need the preceding progression, each term undergoes the division process with the common ratio. Here is an example: 2, 4, 8, 16, 32 …

Here the numbers follow the geometric progression, and the common ratio in this series is 2. The calculation to find GP in any series needs to find the common ratio between the two approximate terms either before or after. In the series that follow this progression type i.e. common ratio is greater than 1, the successor will always be greater than the predecessor. 

This is because the series starts with non-zero terms and undergoes derivation of each consecutive number by the multiplication of a common term. This common term is the common ratio, and the procedure has its general term GP. Here the geometric progression of a series is the sequence that follows the common ratio to get the successor values.

Formula:

Each mathematical calculation has a common way to find the result with variable terms. This is the formula for any method. In geometric progression, the whole sequence follows to find the nth term of the progression. To get the nth term, we need to get two necessary values. If the common ratio is unknown in any series, it is necessary to calculate it by its predecessor. The formula to find the nth term in a sequence is:

an = arn-1

Here ‘a’ is the first term of the geometric progression

‘r’ is the common ratio of the progression and 

‘n’ is the term in the number which we are finding.

Another formula for this type of progression is to find the sum of the N term of GP. This is the calculation to find the sum of all the terms of the geometric progression series. Here the sum of the nth term in the series undergoes the calculation by adding all the terms in the series. 

To find the sum of the n terms, we have to find the nth term and common ratio by the above formula. The nth term provides an idea of the end term, and by substituting values to n in the nth term, we can get the predecessor values of the series. The expression for finding the sum of the n terms can of the geometric progression is:

Sn = a+ar+ar2+ ar3+…+arn-1

And the formula to find the sum is:

Sn = a[(rn-1)/(r-1)] if r ≠ 1

Here the a = the first term of the series

r = the common ratio of the series, which is not equal to 1.

n = number of terms in the geometric progression.

If the common ratio has an equal value with 1 then the expression is a different calculation. The formula for which goes as:

Sn = n*a if r=1

Types of geometric progression:

There are two types of geometric progression in the calculative properties and expression. These types have the categorical basis as the number of terms in the series and the common ratio and end-term values. The two types are:

1. Finite geometric progression.
2. Infinite geometric progression.

• Finite geometric progression:

This type of geometric series consists of a finite number of terms in the expression. This means that in this type of general form of GP, it has a constant value of the term and clearly mentions the last term of expression. For example 1/2, 1/4, 1/8,1/16,……..,1/32768 is the finite type of geometric progression. 

This is because it has a finite value for the last term of expression. This eventually helps in the calculation to find the sum of the N term of GP. The formula and the calculation method of the finite geometric series are distinct as they have the constant value for the last term of the expression. The formula to find the sum of the nth term for this type of series is:

Sn = a[(rn-1)/(r-1)] if r ≠ 1

• Infinite geometric progression:

This type of geometric progression consists of an infinite number of terms in their series. Here in this category, the number of terms does not have a constant value in the progression. Also, here the last term of the series is unknown. The unknown end term is infinite and has a hypothetical value. The general term GP makes this type of expression fall in a special category.

This type of formula to find the sum of the nth value of the geometric progression is the formula of the sum to infinity. The expression of this type of progression goes like a, ar, ar2, ar3 …arn-1...

Conclusion:

The geometric progression is all about the series and its common ratio. The complete topic has the base of calculating the common ratio, which defines the type of expression. This topic has a huge impact on the IIT-JEE syllabus due to its variable scope of questions and formulae.