Recall that a geometric sequence is a sequence in which the ratio of any two consecutive terms is the common ratio, r.
Contents
 1 What is r in series and sequences?
 2 What does r equal in a geometric sequence?
 3 What is nth term in arithmetic sequence?
 4 What is the function of this symbol Σ?
 5 What is the common ratio r?
 6 How do you tell if a sequence is arithmetic or geometric?
 7 How do you determine if a sequence is arithmetic or geometric?
 8 How is r in GP calculated?
 9 What is r in infinite geometric series?
 10 How do you find the nth term in AP?
 11 Arithmetic & Geometric Sequences
 11.0.1 Find the common difference and the next term of the following sequence:
 11.0.2 Find the common ratio and the seventh term of the following sequence:
 11.0.3 Find the tenth term and thenth term of the following sequence:
 11.0.4 Find thenth term and the first three terms of the arithmetic sequence havinga6= 5andd=
 11.0.5 Find thenth term and the first three terms of the arithmetic sequence havinga4= 93anda8= 65.
 11.0.6 Find thenth and the26 th terms of the geometric sequence withanda12= 160.
 12 Geometric Sequences and Series
 13 Summing the First n Terms in a Geometric Sequence
 14 Infinite Geometric Series
 15 Applications of Geometric Series
 16 Series – Mathematics ALevel Revision
 17 Geometric progression – Wikipedia
 18 Elementary properties
 19 Geometric series
 20 Product
 21 History
 22 See also
 23 References
 24 External links
 25 Arithmetico–geometric sequence – Wikipedia
 26 Terms of the sequence
 27 Sum of the terms
 28 Infinite series
 29 References
 30 Geometric Sequences and Sums
 31 Geometric Sequences
 32 The Rule
 33 Why “Geometric” Sequence?
 34 Summing a Geometric Series
 35 Using the Formula
 36 Why Does the Formula Work?
 37 Infinite Geometric Series
 38 Recurring Decimal
 39 Difference between an Arithmetic Sequence and a Geometric Sequence
What is r in series and sequences?
A geometric sequence is an ordered list of numbers in which each term after the first is found by multiplying the previous one by a constant called r, the common ratio.
What does r equal in a geometric sequence?
r is the factor between the terms (called the “common ratio”)
What is nth term in arithmetic sequence?
The nth term of an arithmetic sequence is given by. an = a + (n – 1)d. The number d is called the common difference because any two consecutive terms of an. arithmetic sequence differ by d, and it is found by subtracting any pair of terms an and. an+1.
What is the function of this symbol Σ?
Simple sum The symbol Σ (sigma) is generally used to denote a sum of multiple terms. This symbol is generally accompanied by an index that varies to encompass all terms that must be considered in the sum. For example, the sum of first whole numbers can be represented in the following manner: 1 2 3 ⋯.
What is the common ratio r?
For a geometric sequence or geometric series, the common ratio is the ratio of a term to the previous term. This ratio is usually indicated by the variable r. Example: The geometric series 3, 6, 12, 24, 48,… has common ratio r = 2.
How do you tell if a sequence is arithmetic or geometric?
An arithmetic sequence has a constant difference between each consecutive pair of terms. This is similar to the linear functions that have the form y=mx+b. A geometric sequence has a constant ratio between each pair of consecutive terms.
How do you determine if a sequence is arithmetic or geometric?
If the sequence has a common difference, it’s arithmetic. If it’s got a common ratio, you can bet it’s geometric.
How is r in GP calculated?
Geometric Progression Formulas The nth term from the end of the GP with the last term l and common ratio r = l/ [r(n – 1)].
What is r in infinite geometric series?
An infinite geometric series is the sum of an infinite geometric sequence. The general form of the infinite geometric series is a1+a1r+a1r2+a1r3+, where a1 is the first term and r is the common ratio. We can find the sum of all finite geometric series.
How do you find the nth term in AP?
The formula for the nth term of an AP is, T_{n} = a + (n – 1)d.
Arithmetic & Geometric Sequences
The arithmetic and geometric sequences are the two most straightforward types of sequences to work with. An arithmetic sequence progresses from one term to the next by adding (or removing) the same value on each successive term. For example, the numbers 2, 5, 8, 11, 14,.are arithmetic because each step adds three; while the numbers 7, 3, –1, –5,.are arithmetic because each step subtracts four. The number that is added (or subtracted) at each stage of an arithmetic sequence is referred to as the “common difference”d because if you subtract (that is, if you determine the difference of) subsequent terms, you will always receive this common value as a result of the process.
Below In a geometric sequence, the terms are connected to one another by always multiplying (or dividing) by the same value.
Each step of a geometric sequence is represented by a number that has been multiplied (or divided), which is referred to as the “common ratio.” If you divide (that is, if you determine the ratio of) subsequent terms, you’ll always receive this common value.
Find the common difference and the next term of the following sequence:
3, 11, 19, 27, and 35 are the numbers. In order to get the common difference, I must remove each succeeding pair of terms from the total. There’s no point in choosing which couple I want to sit with as long as they’re right next to one other. To be thorough, I’ll go over each and every subtraction: 819 – 11 = 827 – 19 = 835 – 27 = 819 – 11 = 827 – 19 = 835 – 27 = 819 – 11 = 819 – 11 = 819 – 11 = 819 – 11 = 819 – 11 = 819 – 11 = 819 – 11 = 819 – 11 = 819 – 11 = 819 – 11 = 819 – 11 = 819 – 11 = 819 – 11 Due to the fact that the difference is always 8, the common difference isd=8.
By adding the common difference to the fifth phrase, I can come up with the next word: 35 plus 8 equals 43 Then here’s my response: “common difference: sixhundredandfortiethterm
Find the common ratio and the seventh term of the following sequence:
To get the common ratio, I must divide each succeeding pair of terms by the number of terms in the series. There’s no point in choosing which couple I want to sit with as long as they’re right next to one other. I’ll go over all of the divisions to be thorough: The ratio is always three, hence sor= three. As a result, I have five terms remaining; the sixth term will be the next term, and the seventh will be the term after that. The value of the seventh term will be determined by multiplying the fifth term by the common ratio two times.
When it comes to arithmetic sequences, the common difference isd, and the first terma1is commonly referred to as “a “.
As a result of this pattern, the thenth terma n will take the form: n=a+ (n– 1)d When it comes to geometric sequences, the typical ratio isr, and the first terma1 is commonly referred to as “a “.
This pattern will be followed by a phrase with the following form: a n=ar(n– 1) is equal to a n. Before the next test, make a note of the formulae for the tenth term.
Find the tenth term and thenth term of the following sequence:
, 1, 2, 4, 8, and so forth. Identifying whether sort of sequence this is (arithmetic or geometric) is the first step in solving the problem. As soon as I look at the differences, I see that they are not equal; for example, the difference between the second and first terms is 2 – 1 = 1, while the difference between the third and second terms is 4 – 2 = 2. As a result, this isn’t a logical sequence. As an alternative, the ratios of succeeding terms remain constant. For example, Two plus one equals twentyfour plus two equals twentyeight plus four equals two.
The division, on the other hand, would have produced the exact same result.) The series has a common ratio of 2 and the first term is a.
I can simply insert the following into the formulaa n=ar(n– 1) to obtain the thenth term: So, for example, I may plugn= 10 into the thenth term formula and simplify it as follows_n= 10 Then here’s what I’d say: nth term: tenth term: 256 nth term
Find thenth term and the first three terms of the arithmetic sequence havinga6= 5andd=
The nth term in an arithmetic series has the form n=a+ (n– 1) d, which stands for n=a+ (n– 1) d. In this particular instance, that formula results in me. When I solve this formula for the value of the first term in the sequence, I obtain the resulta= Then:I have the first three terms in the series as a result of this. Because I know the value of the first term and the common difference, I can also develop the expression for the thenth term, which will be easier to remember: In such case, my response is as follows:nth word, first three terms:
Find thenth term and the first three terms of the arithmetic sequence havinga4= 93anda8= 65.
Due to the fact thata4 anda8 are four places apart, I can determine from the definition of an arithmetic sequence that I can go from the fourth term to the eighth term by multiplying the common difference by four times the fourth term; in other words, the definition informs me that a8=a4 + 4 d. I can then use this information to solve for the common differenced: 65 = 93 + 4 d –28 = 4 d –7 = 65 = 93 + 4 d Also, I know that the fourth term is related to the first term by the formulaa4=a+ (4 – 1) d, so I can get the value of the first terma by using the value I just obtained ford and the value I just discovered fora: 93 =a+ 3(–7) =a+ 3(–7) =a+ 3(–7) =a+ 3(–7) =a+ 3(–7) =a+ 3(–7) 93 plus 21 equals 114.
As soon as I know what the first term’s value is and what the value of the common difference is, I can use the plugandchug method to figure out what the first three terms’ values are, as well as the general form of the fourth term: The numbers are as follows: a1= 114, a2= 114– 7, a3= 107– 7, and an= 114 + (n – 1)(–7)= 114 – 7, n+ 7, and an= 121–7, respectively.
Find thenth and the26 th terms of the geometric sequence withanda12= 160.
Due to the fact thata4 anda8 are four places apart, I can determine from the definition of an arithmetic sequence that I can go from the fourth term to the eighth term by multiplying the common difference by four times the fourth term; in other words, the definition tells me that a8=a4+ 4 d. In order to find the common differenced, I need to know what they are. the product of 65 and four days (four days plus four days) is ninetythree and seven days (four days plus seven days). I also know that the fourth term is related to the first term by the formulaa4=a+ (4 – 1) d, so I can find the value of the first terma using the value I just discovered ford and the value I just discovered fora: +3(–7) =a+ 3(–7) =a+ 3(–7) =a+ 3(–7) = a+ 3(–7) = a+ 3(–7) + 21 = a total of 114 As soon as I know what the first term’s value is and what the value of the common difference is, I can use the plugandchug method to figure out what the first three terms’ values are as well as what the general form of the fourth term is: If a1= 114 and a2= 114–7, then 100 is the answer.
If a3= 107–7, then 100 is the answer. If an= (n–1)(n–7), then the answer is (114–7), then the answer is (114–7), then the answer is (121–7) As a result, this is my answer:nth term:121 – 7 nth term:114 – 107, 100
Geometric Sequences and Series
a geometric sequence is an ordered list of integers in which each term after the first is obtained by multiplying the previous one by a constant known asr, the common ratio, and then dividing the result by the preceding one.
Learning Objectives
When the beginning valuea and common ratior are known, calculate the th term of a geometric sequence using those values.
Key Takeaways
 A geometric sequence has the following general form:a, ar, ar2, ar3, ar4, cdots
 A, ar, ar2, ar3, ar4, cdots
 A, ar, ar2, ar3, ar4, cdo In such case, the th term of a geometric sequence with a starting valuenand a common ratioris given by:_ =a
Key Terms
 An ordered list of integers in which each item after the first is found by multiplying the preceding one by a fixed nonzero value known as the common ratio is known as a geometric sequence. A geometric progression is another term for this.
Definition of Geometric Sequences
An ordered list of integers, known as a geometric progression or geometric sequence, in which each item after the first is obtained by multiplying the preceding one by a fixed nonzero number known as the common ratior is defined as follows: Examples of geometric progressions with common ratios are the sequences 2, 6, 18, 54, and cdots. Similarly, the numbers 10, 5, 2.5, 1.25, and cdotsare a geometric sequence with a consistent ratio and presentation style. So the typical shape of a geometric series is:a, ar, ar2/3/4/cdots, ar3, ar4, acdots, and so on.
Behavior of Geometric Sequences
In general, to determine if a particular sequence is geometric, one just examines whether the ratios of the subsequent entries in the series are the same or different. It is possible for the common ratio of a geometric series to be negative, which will result in an alternating sequence. When you have numbers in an alternating sequence, they will alternate between positive and negative signs back and forth. For example, the geometric sequence 1,3,9,27,81,243, cdotsis a series having a common ratio of three.
If the common ratio is as follows:
 Positively, all of the terms will have the same sign as the first term
 Nonetheless, Positive and negative phrases will rotate between each other in the negative case
 More than one will result in exponential growth towards positive infinity (+infty)
 Greater than one will result in exponential growth towards infinity. 1, the progression will consist of a predetermined sequence
 There will be exponential decline toward zero between 1 and 1, but not between 0 and 1. In case of 1, the progression is in the form of an alternating sequence (see alternate series)
 When the absolute values are less than one, there is exponential growth toward positive and negative infinity (due to the alternating sign)
 When the absolute values are more than one, there is exponential growth toward positive and negative infinity (due to the alternating sign)
When the common ratio is not equal to 1,1 or 0, geometric sequences exhibit exponential development or decay, as opposed to the linear increase (or decrease) of an arithmetic progression such as 4, 15, 26, 37, 48, or cdots, the geometric sequences exhibit exponential growth or decay (with common difference11). T.R. Malthus used this finding as the mathematical foundation for his Principle of Population, which he published in 1798. The two types of progression are connected in that exponentiating each term of an arithmetic progression results in a geometric progression, whereas taking the logarithm of each term in a geometric progression with an integer common ratio results in an arithmetic progression.
Summing the First n Terms in a Geometric Sequence
We may sum the terms of a geometric series by employing the common ratio and the first term of the sequence.
Learning Objectives
The total of the firstnterms in a geometric sequence should be calculated.
Key Takeaways
 The terms of a geometric series constitute a geometric progression, which means that the ratio of consecutive terms in the series remains constant over the course of the series. The overall shape of an infinite geometric series is represented by the symbol displaystyle. The behavior of the terms is determined by the common ratior, which is a constant. Forrneq 1, the sum of the firstnterms of a geometric series is provided by the formula displaystyle
 Forrneq 2, the sum of the firstnterms of a geometric series is given by the formula displaystyle
Key Terms
 Geometric series (also known as geometric sequences): An endless sequence of numbers to be added, the terms of which are obtained by multiplying the preceding term by a fixed, nonzero value known as the common ratio
 An infinite sequence of numbers to be added a geometric progression is a series of steps that follows a pattern. When a sequence of numbers is discovered by multiplying the preceding one by a fixed, nonzero value known as the common ratio, the series is said to be infinity.
Although not all geometric series have this quality, geometric series with finite sums are examples of infinite series with finite sums. Geometric series have played an essential part in the development of calculus throughout history, and they continue to be crucial in the study of the convergence of series to the point where they meet. Mathematicians employ geometric series throughout their careers, and they have vital applications in a variety of fields including physics, engineering, biology, economics, computer science, queueing theory, and finance.
For example, the following series:displaystyle+frac+frac+frac+cdots=sum is geometric, since each consecutive term can be generated by multiplying the preceding term by the displaystyle of the previous term in the series.
Each square has an area of frac=frac, with the first square having an area of displaystylecdot frac= frac, and the second square having an area of displaystylecdot frac= frac A number of geometric series with a variety of common ratios are presented in the following.
 The common ratio of 4+40+400+4000+dots is 4+40+400+4000+dots. 10
 The combination of displaystyle+frac+dots has the common ratio
 There is a typical ratio of 3+3+3+3+dots. 1
 Fracfrac+dots+displaystyle+fracfrac+dots+frac+dots+frac+dots+frac+dots+frac+dots+frac+dots+frac+dots+frac+dots+frac+dots+frac+dots+frac+dots+frac+dots+frac+dots+frac+do 33+33+dots has the common ratio of one
 33+33+dotshas the common ratio of three
The value ofrprovides information regarding the nature of the series, which is as follows.
 Because of this, the terms of the series get more smaller and smaller until they reach zero in the limit. Eventually, the series converges to a sum in the interval between 1 and 1. Suppose you have a sequence with the elements (ris onehalf), (frac, (frac, and (cdots right), which has a total of one
 If ris is more than 1 or less than 1, the terms of the series increase in size as ris increases in value. The total of the words increases in size as the series progresses, and the series as a whole has no sum. In the case of Ifris equal to 1, the series diverges, and all the terms of the series are the same. The series begins to diverge
 Ifris1, the terms alternate between two values on the left (text, 2,2,2,2,2,2,cdots right). The sum of the terms oscillates between two values: left and right, respectively (text, 2,0,2,0,2,0,cdots right). A different form of divergence occurs here, and once again, the series has no sum.
Using a formula, we may calculate the total of a finite number of terms included within a sequence. If rneq 1 is true, then the sum of the firstnterms of a geometric series is_displaystyles=a +cdots +a =sum = afracend, wherea is the first term of the series andr is the common ratio. As a result, we may sum the firstnterms by employing the common ratio and the first term of the sequence, as shown in the example.
Example
Find the total of the first five terms of the geometric sequence on the lefthand side of the screen (6, 18, 54, 162, cdots right). When a=6 and n=5, the result is 6. Note also that r = 3, which indicates that each phrase is multiplied by a factor of 3 in order to determine the following term. By substituting these numbers into the sum formula, we get the following result: displaystyles= afrac= 6cdotfrac= 6cdotfrac= displaystyles= afrac= 6cdotfrac= displaystyles the sum of 6 cdot 121 and 726 is 726.
Infinite Geometric Series
Geometric series are one of the simplest instances of infinite series with finite sums, and they are also one of the most common.
Learning Objectives
Understand how to compute the sum of an infinite geometric series and how to detect when a geometric series will converge
Key Takeaways
 For as long as the terms are close to zero, the total of a geometric series is finite
 As the terms go closer to zero, the numbers become insignificantly tiny, allowing the sum to be computed despite the series being infinite. The sum of an infinite geometric series that converges can be computed using the formula displaystyle
 However, this is not recommended.
Key Terms
 Converge on a finite sum: Get close to it. An endless sequence of summed numbers, the terms of which vary gradually with the ratio of the total of the numbers in the sequence
It is an infinite series with terms that follow a geometric development, or with successive terms that have a common ratio, that is known as an infinite geometric series. Whenever the terms of a geometric series approach 0, the total of the terms of the geometric series will be zero. As the numbers go closer to zero, they become insignificantly tiny, which allows a sum to be determined despite the fact that the series is endless. A geometric series with a finite total is said to converge when the sum of the geometric series is zero.
We’ll compute the sums of the following sequence of numbers: displaystyle+frac+frac+cdots frac+frac+ cdots This series shares a common ratio and displaystyle.
By subtracting the new series’ displaystyles from the original series, every term in the original series save the first is null and void: Because displaystylesfracs=1 and s=3 are used, the following code is used: end Any selfsimilar expression may be evaluated using an approach that is comparable to this one.
The following is the formula for summing the firstnterms of any geometric series whererneq 1 is true: The displaystyle s= afrac is defined as follows: If a series converges, we want to discover the sum of not only a finite number of terms, but the total of all of the terms in the series.
right 1, we observe that asnbecomes extremely huge and rnbecomes extremely little.
We may obtain a new formula for the sum of an indefinitely long geometric series by using the rightarrow 0 and applying it: The display styles are afracrightarrow afracquad, textrnrightarrow 0= fracend and textrnrightarrow 0= fracend.
Because of this, forr 1, we may express the infinite sum in the following way: displaystyle
Example
The total of the infinite geometric series must be determined. 64 + 32 + 16 + 8 + cdots = 64 + 32 + 16 + 8 + cdots First and foremost, findr, or the constant ratio between each phrase and the one that comes before it, is as follows: displaystyler= frac= fracend frac= fracend The following substitutions are made into the formula for the sum of an infinite geometric series_a=64and displaystyle r= frac. displaystyles= frac frac frac frac frac frac frac frac frac frac 128 end
Applications of Geometric Series
Mathematical and scientific applications for geometric series may be found in both mathematics and science, and they are one of the simplest instances of infinite series having finite sums.
Learning Objectives
Geometric sequences and series can be used to a variety of physical and mathematical problems.
Key Takeaways
 A repeating decimal can be thought of as a geometric sequence with a common ratio equal to a power of displaystyle
 To compute the area encompassed by a parabola and a straight line, Archimedes utilized the sum of a geometric series as a calculator. The inside of the Koch snowflake is made up of an unlimited number of triangles that are joined together. Geometric series are frequently encountered in the study of fractals as the perimeter, area, or volume of a selfsimilar shape
 In other words, they are fractals in their own right. The understanding of infinite series enables us to resolve old issues such as Zeno’s paradoxes
 And
Key Terms
 Geometric series (also known as geometric sequences): An endless sequence of summed numbers, the terms of which change progressively as a function of a common ratio An example of fractal geometry is a natural event or mathematical set that demonstrates a recurring pattern that may be observed at all scales.
Historically, geometric series played a significant role in the early development of calculus, and they continue to play a vital role in research into the convergence of series today. Mathematicians employ geometric series in a variety of applications. Physics, engineering, biology, economics, computer science (including queueing theory), and finance are just a few of the fields in which they are used extensively. Even while not all geometric series have this attribute, they are one of the most straightforward instances of infinite series with finite sums that can be found.
Repeating Decimal
A repeating decimal can be conceived of as a geometric series whose common ratio is a power of the displaystyle of the numbers in the decimal representation. As an illustration: displaystyle+ frac+ frac+ frac+ frac+ frac+ cdots frac+ frac+ frac+ cdots frac+ frac+ frac+ cdots frac+ frac+ cdots frac+ frac+ cdots frac+ frac+ cdots frac+ frac+ cdots frac+ frac+ The following formula may be used to convert a decimal to a fraction: sum of geometric series formula displaystyle0.7777 cdots= frac= frac= frac= frac= frac= frac= frac= frac= frac= frac= frac= frac= frac= frac= frac= frac= frac= frac= frac= frac= frac= frac= frac= frac= frac= frac= frac= frac= frac= frac= frac= frac The formula is applicable to any repeated word.
Here are a few more illustrations: displaystyle0.123412341234 cdots= frac= frac = fracright)right) = left(fracright)left(fracright)= fracend displaystyle0.09090909 cdots= frac= frac = fracright)right) = left(fracright)left(fracright A repeating decimal with a repeating portion of lengthnis equal to the quotient of the repeating part (as an integer) and10n1 is defined as follows:
Archimedes’ Quadrature of the Parabola
The sum of a geometric series was employed by Archimedes in order to determine the area encompassed by a parabola and a straight line. Using an endless number of triangles, he was able to divide the region into manageable pieces. This is Archimedes’ Theorem, which states that a parabolic segment may be dissected into an endless number of triangles. Archimedes’ Theorem indicates that the entire area beneath the parabola is equal to the area of the blue triangle, which is shown in a different way.
The total area is an endless sequence of right triangles + 4 right triangles + 8 right triangles + cdots assuming that the blue triangle has area1: displaystyle right) +4 right triangles + 8 right triangles + cdots The area of the blue triangle is represented by the first term, the area of the two green triangles is represented by the second term, the area of the four yellow triangles is represented by the third term, and so on.
The following is the result of simplifying the fractions:displaystyle+ frac+ frac+ cdots This is a geometric series with a common ratio of displaystyle, and the fractional component of the series is equal to the common ratio of displaystyle
Fractal Geometry
The interior of a Koch snowflake is made up of an unlimited number of triangles, which is known as the Koch snowflake. There are an endless number of triangles inside the Koch snowflake, making it a fractal form with a fractal interior. Geometric series are frequently encountered in the study of fractals, and they appear as the perimeter, area, or volume of a selfsimilar shape. It is possible to describe the area of a Koch snowflake by means of a geometric series in this example. The first four iterations of constructing the Koch snowflake are as follows: Each iteration adds a set of triangles to the exterior of the form, increasing its overall size.
For example, in the picture above, the triangles added in the second iteration are precisely the size of a side of the largest triangle, and as a result, they have an area that is exactly displaystyle that of the largest triangle.
In terms of area, if we consider the first triangle to be a unit of measurement, the total area of the snowflake is as follows:displaystyleright) +12right) +48right) + cdots The area of the first triangle is represented by the first term of this series, the total area of the three triangles added in the second iteration is represented by the second term of this series, the total area of the twelve triangles added in the third iteration is represented by the third term of this series, and so on.
With the exception of the first term1, this series is geometric in nature, with a constant ratio displaystyle.
Zeno’s Paradoxes
When it comes to philosophy, the Zeno’s Paradoxes are a collection of puzzles designed by an ancient Greek philosopher to promote the notion that truth is in opposition to one’s senses. The following is a simplified version of one of Zeno’s paradoxes: There is a point, A, that desires to go to another point, B. Even if A only moves half of the distance between it and point B at a time, it will never be able to reach point B since you can split the remaining space in half indefinitely. Zeno’s error lies in his assumption that the total of an infinite number of tiny steps cannot be greater than one hundred thousand.
By looking at the convergence of the geometric series with the displaystyle, we can see that his paradox is not valid. With the help of contemporary mathematics, this challenge has been resolved, since the notion of infinite series can be used to calculate the total of the distances traveled.
Series – Mathematics ALevel Revision
In a sequence, the series is the sum of the sequence up to a specific number of entries. It is frequently abbreviated as S n. The sum of three terms equals S 3 = 2 + 4 + 6 = 12 if the sequence is 2, 4, 6, 8, 10, and so on. The Sigma Notation is a mathematical notation that represents the relationship between two numbers. It is common to use the Greek letter sigma, abbreviated S, to denote the sum of a sequence of numbers. This is best described with the help of an example: This suggests that you should replace the r in the phrase with 1 and write down the result.
 Continue repeating this until you reach the number 4, which is the number above the letter S.
 So the sum is 31 plus 32 plus 33 plus 34, which is three plus six plus nine and twelve, which is thirty.
 The Speculative Case n SU rr = 1 SU rr = 1 SU rr = 1 This is the most common scenario.
 In the above example, U r= 3r + 2 and n = 3 are the values.
 An arithmetic progression is a series in which each phrase is a specific number greater than the term before it in the sequence.
 For example, the arithmetic progression 3, 5, 7, 9, 11 is an arithmetic progression when d = 2.
 In general, the nth term of an arithmetic progression with a first term and a common difference is: a + (n – 1)d for the nth term of the progression.
We already knew that U n= 3 + 2(n – 1) = 2n + 1, so we didn’t have to think about it.
It is derived in the following way: The sum of n terms can be calculated as follows: S n= a + (a + d) + (a + 2d) +.
+ (a + (n – 1)d) (1) If we reverse the order of these words, we get: The number S n is equal to (a + (n – 1)d) + (a + (n – 2)d) +.
+ S n=++.
+ S n=++.
the first 20 odd numbers).
The common ratio of the series is denoted by the letter r. There are two ways to express the nth term of geometric progression, where a represents the first term and r represents the common ratio. The first is as follows:
The following geometric progression, for example, begins with 1, and the typical ratio is 2:1, then 2, 4, 8, 16, and so on till the end. As a result, the nth term equals 2 n1. geometric progression is the total of all geometric progressions. The sum of the first n terms of a geometric progression is represented by the expression: This may be demonstrated in the following ways: The number S n equals a + ar + ar 2+. + ar n1 (1) When we multiply by r:rS n= ar + ar 2+. + ar n(2)(1) – (2), we get the following result: S n (1 – r) = a – ar n = a – ar n (since all the other terms cancel) In this case, dividing through by 1 – r gives us the formula shown in this section.
 S5 = 2 is a mathematical expression.
 If r1 is less than 1 and bigger than –1 (in other words, whenever r is less than 1 and higher than –1), the total of the series as n approaches infinity approaches a certain number.
 This value is equal to the following: Example Calculate the sum of the following sequences up to infinity: In this case, a = 1/2 and r = 1/2 Because of this, 0.5/0.5=1 represents the total to infinity.
 A Difficult Case Study When it comes to arithmetic progressions, the first, second, and fifth terms of an arithmetic progression correspond to the first three terms of a geometric progression.
 Find the two possible values for the fourth term of the geometric progression in the following manner: The initial term in the arithmetic progression is represented by the symbol a.
 The fifth term is as follows: a + 4d As a result, the first three terms of the geometric progression are denoted by the letters a, a + d, and a + 4d.
 As a result, the relationship between the second term and the first term is the same as the relationship between the third term and the second term.
 As a result, if d = 0, then r = 1.
 In the arithmetic progression, we are told that the third term is the number 5.
 As a result, when d = 0 and a = 5, and when d = 2a and a = 1, a = 1.
 As a result, when d = 0, a = 5, and r = 1, the result is 1.
In this situation, the geometric progression is 5, 5, 5, 5, 5, and hence the fourth term is 5 in the progression. Assuming that d = 2a, r = 3, and a = 1, the geometric progression is as follows: 1, 3, 9, 27, and the fourth term is as follows: 27.
Geometric progression – Wikipedia
Diagram exhibiting three fundamental geometric sequences of the pattern 1(rn 1), each of which may be repeated up to six times. This is a unit block, and the dashed line symbolizes the infinite sum of the series, a number that it will eternally approach but never reach: 2, 3/2, and 4/3 in the case of the first block. According to mathematics, ageometric progression (also known as ageometric sequence) is a series of nonzero integers in which each term after the first is determined by multiplying the preceding one by a fixed, nonzero number known as the common ratio In this case, the sequences 2, 6, 18, 54, and so on are all geometric progressions with a common ratio of 3.
is a sequence having a common ratio of 1/2.
This is the general form of a geometric series, in whichr0 denotes the common ratio.
The difference between a progression and a series is that a progression is a sequence, but a series is a total of the elements in the progression.
Elementary properties
Then the th term of a geometric series with starting valuea=a1 and common ratioris represented by the expression. A geometric sequence of this type also follows the recursive relation for every number it contains. In general, to determine if a particular sequence is geometric, one just examines whether the ratios of the subsequent entries in the series are the same or different. In some geometric sequences, the common ratio might be negative, resulting in an alternating sequence in which the integers alternate between being positive and being negative.
The behavior of a geometric series is dictated by the value of the common ratio in that sequence.
 Positive phrases will all have the same sign as the beginning term
 Negative terms will rotate between positive and negative signs
 And neutral terms will have no sign. greater than 1, there will be exponential growth towards positive or negative infinity (depending on the sign of the initial term)
 Greater than 1, the progression is a constant sequence
 Between 1 and 1 but not zero, there will be exponential decay towards zero ( 0)
 Less than 1, for the absolute values of each term in the sequence, there will be exponential growth towards (unsigned) infinity, due to the alternating sign
 Less than 1, the progression is a constant sequence
 Greater
Exponential growth or exponential decay can be observed in geometric sequences (with common ratios other than 1, 1 or 0), as opposed to linear growth (or decrease) shown in anarithmetic progression such as 4, 15, 26, 37, 48,. (with commondifference11). T.R. Malthus used this finding as the mathematical foundation for hisPrinciple of Population, which he published in 1798. The two types of progression are connected in that exponentiating each term of an arithmetic progression results in a geometric progression, and calculating the logarithmof each term in a geometric progression with a positive common ratio results in an arithmetic progression.
Geometric series
2  +  10  +  50  +  250  =  312  
− (  10  +  50  +  250  +  1250  =  5 × 312)  


2  −  1250  =  (1 − 5) × 312 
Calculation of the sum of two numbers plus ten numbers plus fifty numbers plus 250 numbers. In order to deduct from the original sequence, the sequence is multiplied by 5 for each word in the sequence. In the end, there are just two terms left: the first term (a), and the term one beyond the last (orarm). 312 is obtained by subtracting these two components from one another and dividing the result by onefifth. The sum of the numbers in a geometric progression is known as an ageometric series.
For example, because the derivation (below) does not rely on the fact that a and r are real numbers, it is valid for complex numbers as well.
Derivation
In order to obtain this formula, first write a generic geometric series in the following format: Adding 1r to both sides of the previous equation yields a simpler formula for this sum, and we’ll see that because all the other terms cancel, this is the simplest formula we can come up with for this sum. With respect to r 1, we may rearrange the preceding to obtain the easy formula for a geometric series that computes the sum of n terms: If r 1 = 1, we have: If one were to start the sum from a different number, saym, rather than from k=1, the result would be supplied.
This formula may be transformed into a sum of the form by differentiating it with regard to.
For a series with only a few wacky abilities of sleight of hand, When the Stirling numbers of the second sort are added together, an accurate formula for the generalized sum is obtained.
Infinite geometric series
An infinite geometric series is a series that has no end. is an infinite series in which all of the terms have the same ratio between them. If and only if the absolute value of the common ratio is smaller than one (r1), then a series of this type will converge. Using the finite sum formula, it is possible to determine its value. Animation demonstrating the convergence of partial sums of geometric progression (red line) to the sum of geometric progression (blue line). Diagram illustrating the geometric sequence 1 + 1/2 + 1/4 + 1/8 +, which converges to the number 2.
If the norm ofris smaller than one, the latter formula is true in everyBanach algebra, and it is also valid in the field ofp adic numbers, provided rp1 is less than one.
For example, this formula is only valid for r1 and not for r2.
It is an ageometric series with a first term of 1/2 and a common ratio of 1/2, hence the total of its terms is 1/2.
The inverse of the above series is 1/2 1/4 + 1/8 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1 is a straightforward illustration of an alternating series that converges absolutely. It is an ageometric series with a first term of 1/2 and a common ratio of 1/2, hence the total of its terms is 1/2.
Complex numbers
The summation formula for geometric series is still valid even when the common ratio is a complex number, as seen in the following example. In this situation, the requirement that the absolute value ofrbe less than 1 is replaced with the condition that themodulusofrbe smaller than one. It is feasible to compute the sums of several geometric series that are not immediately clear. Take, for example, the following proposition: That this is the case may be demonstrated by the fact thatEuler’s formula has an unintended effect The result of including this into the original series is.
Product
The product of a geometric progression is equal to the sum of all terms in the progression. In a nutshell, it may be easily computed by taking the geometric mean of the progression’s first and final individual terms and increasing that mean to the power of the number of terms. The formula for the sum of terms in anarithmetic series is essentially similar: take the arithmetic mean of the first and final individual terms, then multiply by the number of terms in the sequence Due to the fact that the geometric mean of two integers equals the square root of their product, the product of a geometric progression is as follows: It’s worth noting that, although though this formula requires taking the square root of a possibly odd power of a potentially negativer, it will not give a complicated result as long as neitheranorrhas an imaginary component.
A negative intermediate result can be squared to produce an imaginary number if the result is negative and odd.
A hypothetical intermediate created in this manner, on the other hand, will be raised to the power of, which must be an even number becausenby itself was an odd number; hence, the ultimate result of the computation may possibly be an odd number, but it could never be an imagined one.
Proof
Let’s say you want to present a product. It can only be calculated by explicitly multiplying each individual term together, according to the definition. Written out in its entirety, Performing the multiplications and gatherings in the same manner as words, The sum of an arithmetic sequence is represented by the exponent ofr. Substituting the formula for that computation allows for the equation to be simplified, which is advantageous. Rewritingaas,which brings the proof to a close.
History
MS 3047, a clay tablet from Mesopotamia’s Early Dynastic Period that depicts a geometric progression with base 3 and multiplier 1/2, has been discovered. It has been hypothesized that it is Sumerian in origin, and that it comes from the city ofShuruppak. It is the first known example of a geometric progression dating back to before the period when Babylonian mathematics was developed.
Euclid’sElements, books VIII and IX, investigate geometric progressions (such as thepowers of two; see the page for more information) and provide some of their characteristics, which are summarized in this article.
See also
 Arithmetic progression is the order of numbers in a sequence. Arithmeticogeometric sequence
 Linear difference equation
 Arithmeticogeometric sequence The exponential function is a mathematical function that belongs to a certain class. The evolution of harmony
 Harmonic series are infinite series of the reciprocals of the positive integers
 They are also known as harmonic series. Infinite series–infinite sum is a mathematical concept. Recommended number– Standard rules for selecting specific product dimensions within the restrictions of a particular set of requirements. Thomas Robert Malthus was a British political economist who lived from 1766 to 1834. Geometric distribution and probability distribution are two different things.
References
 The book Higher Algebra by HallKnight (ISBN8181160002) has a page number of 39.
External links
 Geometric progression is covered in the Encyclopedia of Mathematics, published by EMS Press in 2001. atMathalino.com, formulae for the sum of finite and infinite geometric progression were deduced and published. a Geometric Progression Calculator, as well as a visually appealing demonstration of a geometric progression Sumatsputsoft.com
 Weisstein, Eric W., “Geometric Series,” MathWorld
Arithmetico–geometric sequence – Wikipedia
arithmetico–geometric sequence is a type of sequence in mathematics that is formed by multiplying the terms of ageometric progression with the equivalent terms of anarithmetic progression term by term. To put it another way, the product of the then th term of an arithmetico–geometric sequence and the then th term of a geometric sequence is the then th term of an arithmetico–geometric sequence. Numerous applications of arithmetico–geometric sequences, such as the computation of anticipated values in probability theory, may be traced back to them.
Specifically, the arithmetic component appears in the numerator (in blue), whereas the geometric component appears in the denominator (in red) (in green).
Linear difference equations are a specific instance of the sequences described above.
Terms of the sequence
In an arithmetico–geometric sequence comprised of anarithmetic progression(in blue) with difference and initial value and ageometric progression(in green) with initial value and common ratio, the first few terms may be expressed as:
Example
For example, the sequence is defined by the words and.
Sum of the terms
In an arithmetico–geometric sequence, the sum of the firstnterms has the formwhereandare the firstnterms in both the arithmetico–geometric sequence and the geometric sequence, respectively. This sum is represented by the closedform equation.
Proof
By multiplying,byr, we get The sum of a geometric series may be calculated by subtractingrS n fromS n and applying the concept of telescoping series to obtainwherethe final equality arises from the statement for the sum of a geometric series The result is obtained by dividing through by1 r at the end.
Infinite series
The sumSof the arithmetico–geometricseries, that is, the sum of all the infinitely many terms of the progression, is given byIf lies outside of the aforementioned range, the series either has a zero sum or has an infinite number of terms.
 Diverges(whenr1 or whenr= 1 where the series is arithmetic andaanddare not both zero
 If bothaanddare zero in the latter case, all terms of the series are zero and the series is constant)
 Diverges(whenr1 or whenr= 1 where the series is arithmetic andaanddare not both zero)
 Oralternates(whenr1)
 Oralternates(whenr1)
Example: application to expected values
The sum of an arithmetico–geometric series defined by and converges to, for example. This sequence relates to the amount of coin tosses that should be expected before receiving “tails.” It is possible to get tails for the first time at the kth toss if the following conditions are met: As a result, the predicted number of tosses may be calculated as
References
 In D. Khattar’s The Pearson Guide to Mathematics for the IITJEE, 2nd edition (New Edition). P. Gupta, P. Pearson Education India, p. 10.8, ISBN 8131728765
 Pearson Education India. Isbn: 9788170085977
 Comprehensive Mathematics XI, Laxmi Publications, page 380, ISBN: 9788170085977
Geometric Sequences and Sums
A sequence is a collection of items (typically numbers) that are arranged in a specific order.
Geometric Sequences
Geometric Sequences are formed by multiplying the terms of the preceding sequence by a constant.
Example:
1, 2, 4, 8, 16, 32, 64, 128, 256, are all possible combinations.
Each number in this series is separated by a factor of two. Each phrase (with the exception of the first term) is discovered by multiplying the preceding term by two times. In general, we write a Geometric Sequence in the following format:where:
 A is the first term, while r is the factor that separates the two terms (also known as the “common ratio”).
Example:
The series begins at 1 and increases by a factor of two each time, thus
 A=1 (the first term)
 R=2 (the “common ratio” between terms is increased by a factor of two)
As a result, we obtain:= =But be careful,rshould not be 0:
 In the case of r=0, we get a sequence that is not geometric.
The Rule
We may also compute any term by applying the following rule: x n= ar x n= ar (n1) (We use the notation “n1” since the first term is equal to zero.)
Example:
10, 30, 90, 270, 810, and 2430 are the numbers to remember. Each number in this series is separated by a factor of three. The following are the values ofa andrare: The Rule is as follows for any term:x n= 10 3 (n1) As a result, the fourth term is as follows:x 4= 10 3 (41)= 10 3 3= 10 27 = 270 In addition, the tenth term is as follows:x ten = 10 x 3 (101)= 10 x 3 9= 10 x 19683 = 196830A It is also possible for Geometric Sequence to have smaller and smaller values.
Example:
The interval between each number in this series is divided by 0.5 (half). Its Rule is as follows: x n= 4 (0.5) n1
Why “Geometric” Sequence?
Because it is equivalent to expanding the ingeometry’s dimensions:
a line is 1dimensional and has a length ofr  
in 2 dimensions a square has an area ofr 2  
in 3 dimensions a cube has volumer 3  
etc (yes we can have 4 and more dimensions in mathematics). 
Geometric Sequences (also known as Geometric Progressions (G.P.’s)) are a type of geometric progression.
Summing a Geometric Series
To summarize, a + ar + ar 2+ = a + ar + ar 2+. + ar (n1) The number of terms in each term isar, where k starts at 0 and continues all the way up to n1). We may make use of the following useful formula: a is the first of two terms r is the “common ratio” between phrases, and n is the number of terms in the sentence. What exactly is that amusing symbol? It is referred to as The Sigma Notation is a type of notation that is used to represent a sigma function. Additionally, the starting and finishing values are displayed below and above it: “Sum upnwherengoes from 1 to 4,” the text states.
The formula is straightforward to use.
Example: Sum the first 4 terms of
10, 30, 90, 270, 810, and 2430 are the numbers to remember. Each number in this series is separated by a factor of three. The values ofa,randnare as follows:
 A = 10 (the first term)
 R = 3 (the “common ratio”)
 N = 4 (we want to add the first four terms together)
 A = 10 (the first term).
So:Becomes: You may doublecheck it yourself by going to: 400 is equal to 10 plus 30 plus 90 plus 270. And, certainly, because there are only four terms in this case, it is simpler to simply add themin. Consider, however, the addition of 50 words. The formula becomes considerably simpler after that.
Using the Formula
Let’s see how the formula works in practice:
Example: Grains of Rice on a Chess Board
On the pageBinary Digits, we show an illustration of rice grains on a chess board, which you can see here. The following question is posed: When we lay rice on a chess board, we are essentially saying:
 There will be one grain on the first square, two grains on the second square, four grains on the third and so on.
How many grains of rice are there in total if you double the grains of rice on each square? So we have the following:
 A = 1 (the first term)
 R = 2 (each term doubles)
 N = 64 (the number of squares on a chess board)
As a result, the equation becomes:=12 64 1= 2 64 1 = 18,446,744,073,709,551,615 Exactly the outcome we were looking for on theBinary Digitspage (thank heavens!) was obtained. And here’s another example, this time with a number lower than 1:
Example: Add up the first 10 terms of the Geometric Sequence that halves each time:
The values ofa,randnare as follows:
 A = 12 (the initial term)
 R = 12 (each time in half)
 N = 10 (total of 10 words to add)
As a result, it becomes:Very near to 1. (Question: what happens if we continue to raise the amount?)
Why Does the Formula Work?
Let’s take a look at why the formula works because we’ll be employing an unusual “technique” that’s worth understanding. To begin, we’ll refer to the entire total as “S”: S = a + ar + ar 2+. + ar (n2)+ ar (n1) + ar (n2)+ ar (n1) After that, multiplySbyr: Sr = ar + ar 2+ ar 3+. + ar (n1)+ ar n. + ar (n1)+ ar n. Have you noticed howSandSrare andSandSrare are similar? Nowsubtractthem! Wow! All of the phrases in the center are perfectly balanced. In this case, it’s a clever trick.
By subtractingSr fromS, we get We get a straightforward result: S r = a r n S r = a r n Let’s restructure it such that we can findS: Sanda: Take this into consideration. S(1 r) = a(1 r n) = a(1 r n) Divide by (1r):S =a(1 r n)(1 r)(1 r)(1 r)(1 r) Here’s our recipe (tada!) for success:
Infinite Geometric Series
So, what happens when the number reaches infinity? We can use the following formula: However, use caution: The value of r must be between (but not including) 1 and 1, and the value of r should not be 0 because the sequence is not geometric in nature. Consequently, when the ratio is less than 1 (and more than 1), our infnite geometric series has an infinite total. Let’s go back to our earlier example and see what occurs this time around:
Example: Add up ALL theterms of the Geometric Sequence that halves each time:
As a result, the equation is:=12112= 1Yes, adding 12+14+18+. etc is equivalent to precisely one.
Don’t believe me? Just look at this square:By adding up12+14+18+. we end up with the whole thing! 
Recurring Decimal
On another page, we raised the question “Does 0.999. equal 1?”. Let’s see if we can figure out how to compute it:
Example: Calculate 0.999.
We may express a repeating decimal as a sum in the following way: And now we can put the formula to good use: Yes! 0.999 is the same as 1. That’s all there is to it. Geometric Sequences (and their sums) are capable of performing a wide range of astonishing and powerful operations.
Difference between an Arithmetic Sequence and a Geometric Sequence
Numerical systems and associated operations are dealt with in arithmetic, which is a type of mathematical process. It is employed in order to get a single, definite value. The term “arithmetic” is derived from the Greek word “arithmos,” which literally translates as “numbers.” It is a branch of mathematics that focuses on the study of numbers and the characteristics of basic operations such as addition, subtraction, multiplication, and division. It is a subject of mathematics that is primarily concerned with the study of numbers.
The two most prominent forms of mathematical sequences are arithmetic and geometric sequences, respectively.
An alternative type of sequence is a geometric sequence, which has a predetermined ratio between each pair of successive words.
Arithmetic Sequence
The phrase Arithmetic Sequence refers to a sequence of integers in which the difference between any two successive terms is always the same, regardless of how many terms are in the sequence. To put it another way, it signifies that the next number in the series is computed by multiplying the previous number by a predetermined integer. An Arithmetic Sequence can also be written asa, a + d, a + 2d, a + 3d, a + 4d, where a = the first term and d = the last term. d is the most common difference between two words in a sentence.
The constant difference is 6 in each of the numbers from 5 to 11.
Geometric Sequence
The phrase Geometric Sequence refers to a sequence of integers in which the ratio of any two successive terms is always the same, regardless of the terms’ order. To put it another way, it implies that the next number in the series is computed by multiplying a predetermined number by the number that came before.
Furthermore, a Geometric Sequence may be represented as:a, ar, ar 2, ar 3, ar 4,.a, ar, ar 2, ar 3, ar 4,. where a denotes the first term and d is the common difference between terms. For example, the numbers 2, 6, 18, 54, 162,. In this instance, the constant multiplier is three.
How can you tell the difference between an Arithmetic sequence and a Geometric sequence?
The following considerations are critical in distinguishing between arithmetic and geometric sequences, respectively:
 An arithmetic Sequence is a collection of numbers in which each new phrase changes from the preceding term by a predetermined percentage of the total number of terms. It is a sequence of numbers in which each element after the first is created by multiplying the previous number by a constant factor
 It is also known as the Geometric Sequence. When there is a common difference between successive terms, denoted by the letter ‘d,’ a series can be characterized as arithmetic. When there is a common ratio between following phrases, as represented by the letter ‘r,’ the sequence is said to be geometric. When a new term is introduced into an arithmetic sequence, it is created by adding or subtracting a fixed value from the preceding term. The next term is found by multiplying or dividing a fixed value from the preceding term, as opposed to geometric sequence, which uses a variable value. It is linear in nature that the variance between elements of an arithmetic series occurs. The variance in the elements of the sequence, on the other hand, is exponential. According to the context, infinite arithmetic sequences diverge, but infinite geometric sequences either converge or diverge, depending on the situation.
There is a distinction between an arithmetic sequence and a geometric sequence.
S.No.  Arithmetic sequence  Geometric sequence 

1  Arithmetic Sequence is a set of numbers in which each new phrase differs from the previous term by a fixed amount.  A geometric sequence is a collection of integers in which each subsequent element is created by multiplying the previous number by a constant factor. 
2  Between successive words, there is a common difference.  Between successive words, they have the same common ratio. 
3  Subtraction or addition are used to get terms.  Division or Multiplication are used to get terms. 
4  Example: 5, 11, 17, 23, 29, 35,…  Example: 2, 6, 18, 54, 162,… 
Sample Problems
The first question is: What exactly is a Geometric Sequence, and why is it thus referred to as such? Answer:A geometric sequence is a collection of integers that are linked together by dividing or multiplying by a value that is comparable to one another. Question 2: Is it possible for an Arithmetic Sequence to be both geometric and arithmetic in nature? The answer is that in mathematics, an arithmetic sequence is defined as a series in which the common difference, also known as the variance, between successive integers remains constant.
Thus, a series cannot be both geometric and arithmetic at the same time due to this constraint.
When the difference between two following members of the series equals a constant term, we have an arithmetic sequence.
In order to determine the n thterm in an arithmetic series, what is the technique to follow?
A common difference of an arithmetic series is represented by the first term, ‘a,’ and the common difference of the second term, ‘d,’ In order to determine the nthterm of a geometric series, you must first determine the nthterm of the sequence.