A geometric progression is a sequencewhere every term bears a constant ratioto its preceding term. Geometric progression is a special type of sequence. In order to get the next term in the geometric progression, we have to multiply with afixed term known as the common ratio,every time, and if we want to find the preceding term in the sequence, we just have to divide the term with the same common ratio. Here is anexample of a geometric progression is 2, 4, 8, 16, 32, …… having a common ratio of 2.
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The geometric progressions can be a finite series or an infinite series. The common ratio of a geometric progression can be a negative or a positive integer. Here we shall learn more about the geometric progression formulas, and the different types of geometric progressions.
| 1. | Geometric Progression Introduction |
| 2. | Geometric Progression Formula |
| 3. | Geometric Progression Sum Formula |
| 4. | Geometric Progression Examples |
| 5. | Practice Questions on Geometric Progression |
| 6. | FAQs on Geometric Progression |
Geometric Progression Introduction
A geometric progression is a special type of sequencewhere the successive terms bear a constant ratio know as a common ratio. Geometric progression is also known as GP. The geometric sequence is generally represented in form a, ar, ar2…. where a is the first term and r is the common ratioof the sequence. The common ratio can have both negative as well as positive values.To find the terms of a geometric series, we only need the first termand the constant ratio.
The geometric progression isof two types. The two types of geometric progressions are based on the number of terms in the progression series. The two types of a geometric progression are the finite geometric progression and the infinite geometric progression. The details of the two geometric progressions are as follows.
Finite geometric progression
Finite geometric progression is the geometric series that contains a finite number of terms. It is the sequence where the last term is defined. For example 1/2,1/4,1/8,1/16,…,1/32768 is a finite geometric series where the last term is 1/32768.
Infinite geometric progression
Infinite geometric progression is the geometric series that contains an infinite number of terms. It is the sequence where the last term is not defined. For example, 3, −6, +12, −24, +… is an infinite series where the last term is not defined.
Geometric Progression Formula
The geometric progression formula is used to find the nth term in the sequence. To find the nth term in the geometric progression, we require the first term and the common ratio. If the common ratio is not known, the common ratio is calculated by finding the ratio of any term by its preceding term.The formula for the nth term of the geometric progression is:
(a_n) = arn-1
where
a is the first termr is the common ration is the number of the term which we want to find.
Geometric Progression Sum Formula
The geometric progression sum formula is used to find the sum of all the terms in a geometric sequence. As we read in the above section that geometric sequenceis of two types, finite and infinite geometric sequences, hence the sumof their termsis also calculated by different formulas.
Finite Geometric Series
If the number of terms in a geometric sequence is finite, then the sum of the geometric series is calculated by the formula:
(S_n) = a(1−rn)/(1−r) forr≠1, and
(S_n) = an for r = 1
where
a is the first termr is the common ratio n is the number of the terms in the series
Infinite Geometric Series
If the number of terms in a geometric sequence is infinite,an infinite geometricseries sum formula is used. In infinite series, there arise two cases depending upon the value of r. Let us discuss the infinite series sum formula for the two cases.
Case 1: When|r| a is the first termr is the common ratio
Case 2:|r| >1
In this case, the series does not converge and it has no sum.
Geometric Progression vs Arithmetic Progression
Here are a few differences between geometric progression and arithmetic progression shown in the table below:
| GP has the same common ratio throughout.
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AP does not have common ratio. |
| GP does not have common difference. | AP has the same common difference throughout. |
| A new term is the product of the previous term and the common ratio | A new term is the addition of the previous term and the common difference. |
| An infinite geometric sequence is either divergent or convergent. | An infinite arithmetic sequence is divergent. |
| The variation of the terms is non-linear. | The variation of the terms is linear. |
Important Notes on Geometric Progression
In a geometric progression, each successive term is obtained by multiplying the common ratio to its preceding term.
The formula for the nthterm of a geometric progression whose first term is aand common ratio is (r) is: (a_n=ar^{n-1})
The sum of n terms in GP whose first term is aand the common ratio is rcan be calculated usingthe formula: (S_n=dfrac{a(1-r^n)}{1-r})
The sum of infinite GP formula is given as: (S_n=dfrac{a}{1-r}) where |r|
Related Topics on Geometric Progression
Check out these interesting articles related to geometric progression:
Observe that each square is half of the size of the square next to it. Which sequence does this pattern represent?
Solution:
Let's write the geometric progression seriesrepresented in the figure.
1, 1/2, 1/4, 1/8 …
Every successive term is obtained by dividing its preceding term by 2
The sequence exhibits a common ratio of 1/2.
Answer:The pattern represents the geometric progression.
Question 2: In a certain culture, the count of bacteria gets doubled after every hour. There were 3 bacteria in the culture initially. What would be the total count of bacteria at the end of the 6th hour?
Solution
Here, the number of bacteria forms a geometric progression where the first term ais 3 and the common ratio ris 2.
So, the total number of bacteria at the end of the 6th hour will be the sum of the first 6 terms of this progression given by (S_6).
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(S_6) = 3(26−1)/(2−1)
=3(64−1)/1
=3×63
=189
Answer:So, the total count of bacteria at the end of the 6th hour will be 189.
Example 3: Find the following sum of the terms of this infinite geometric progression:
