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Sum of an Infinite Arithmetic Geometric Series

Sum of an Infinite Arithmetic Geometric Series

Edited By Komal Miglani | Updated on Jul 02, 2025 07:32 PM IST

Arithmetico- Geometric series is a series with a combination of the Arithmetic series and Geometric series. This series is formed by taking the product of the corresponding elements of arithmetic and geometric progressions. In short form, it is written as A.G.P (Arithmetico-Geometric Progression). In real life, we use the sum of an Infinite Arithmetic Geometric Series for analyzing current flow and sound waves.

This Story also Contains
  1. Arithmetico-Geometric Series
  2. The sum of an Infinite Arithmetico Geometric Series
  3. Solved Examples Based on Arithmetico-Geometric Series
Sum of an Infinite Arithmetic Geometric Series
Sum of an Infinite Arithmetic Geometric Series

In this article, we will cover the concept of the Sum of an Infinite Arithmetic Geometric Series. This category falls under the broader category of Sequence and series, which is a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination(JEE Main) and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Questions based on this topic have been asked frequently in JEE Mains.

Background wave

Arithmetico-Geometric Series

Arithmetico-geometric series is the combination of arithmetic and geometric series. This series is formed by taking the product of the corresponding elements of arithmetic and geometric progressions. In short form, it is written as A.G.P (Arithmetico-Geometric Progression).

Let the given AP be a,(a+d),(a+2d),(a+3d),
And, the GP is 1,r,r2,r3,
Multiplying the corresponding elements of the above progression, we get, a,(a+d)r,(a+2d)r2,(a+3d)r3,
This is a standard Arithmetico-Geometric Progression.
Eg: 1,3x,5x2,7x3,9x4,

The sum of an Infinite Arithmetico Geometric Series

S denotes the sum of an infinite AGP. This sum is a finite quantity if -1 < r < 1

S=a+(a+d)r+(a+2d)r2+(a+3d)r3

Multiply both side of eq (i) by 'r '
r S=ar+(a+d)r2+(a+2d)r3+(a+3d)r4.

Subtract eq (ii) from eq (i)
(1r)S=a+(dr+dr2+dr3+.. upto )(1r)S=a+dr1rS=a1r+dr(1r)2

The sum of infinite AGP is given by
S=a1r+dr(1r)2
here |r|<1

Where,

a= first term of AGP

d= common difference of AP

r= Common ratio of GP

Recommended Video Based on Sum of an Infinite Arithmetico Geometric Progression:

Solved Examples Based on Arithmetico-Geometric Series

Example 1: If the sum of the series (1213)+(122123+132)+(1231223+1232135)+(1241223+1223212232+134)+ is αβ
, where α and β are co-prime, then α+3β is equal to [JEE MAINS 2023]

Solution: The sum of the series represents Arithmetico - Geometric Progression,
P=(1213)+(122123+132)+(123+1223+1232132)+P(12+13)=(122132)+(123+133)+(124134)+5P6=16112181+135P6=12112512P=12=αβα=1,β=2α+3B=7

Hence, the required answer is 7.

Example 2: Suppose a1,a2,2,a3,a4 be in an arithemetico-geometric progression. If the common ratio of the corresponding geometric progression is 2 and the sum of all 5 terms of the arithmetico-geometri 49 progression is 492, then a4 is equal to [JEE MAINS 2023]

Solution:
Given,
Common ratio =r=2

The sum of all 5 terms of the arithmetico- geometric progression is 492
(a2d)4,(ad)2,a,2(a+d),4(a+2d)a=2(14+12+1+6)×2+(1+2+8)d=4922(34+7)+9d=4929d=492624=98624=9d=1a4=4(a+2d)=16

Hence, the required answer is 16.

Example 3: Let a1,a2,a3, be an A.P. If r=1ar2r=4, then 4a2
is equal to [JEE MAINS 2022]
Solution
b=s=a12+a222+a323+s2=a422+a223+s2=a12+d(122+123+s2=a12+d(14112)s=a1+d1=a2=4 or 4a2=16

Hence, the required answer is 16.

Example 4: If 6312+10311+20310+4039++102403=2nm, where m is odd, then m.n is equal to [JEE MAINS 2022]
Solution: The given series represents Arithmetico- Geometric Progression,
6312+10311+20310++1024031=1312+(5312+10311++1024031)
G.p. with r=2×3=6,n=12
=1312+5312(612161)=1312+6121312=612312=212m=1,n=12mn=12

Hence, the answer is the 12.

Example 5: Let S=2+67+1272+2073+3074+ Then 4 S is equal to
Solution: The given series represents Arithmetico-Geometric Progression, [JEE MAINS 2022]
S=2+67+1272+2073+3074+S7=27+672+1273+2074+ Subtract 6 S7=2+47+672+873+1074+672 S=27+472+673+874+

 Subtract 67(117)S=2+27+272+273+274+3649 S=2(111/7)3649 S=73S=7322334 S=(73)3

Hence, the required answer is (73)3


Frequently Asked Questions (FAQs)

1. What is Arithmetic - Geometric Series?

 Arithmetico-geometric series is the combination of arithmetic and geometric series. This series is formed by taking the product of the corresponding elements of arithmetic and geometric series.

2. What is the sum of infinite Arithmetic-Geometric series?

The sum of infinite Arithmetic-Geometric series represents the sum of infinite terms of Arithmetic-Geometric series.

3. What is the difference between Arithmetic and Geometric Progression?

An arithmetic progression is a sequence in which each term increases or decreases by a constant term or fixed number. This fixed number is called the common difference of an AP and is generally denoted by ‘d’. The nth term (general term) of the A.P. is an=a+(n1)d  . 

whereas, A geometric sequence is a sequence where the first term is non-zero and the ratio between consecutive terms is always constant. The ‘constant factor’ is called the common ratio and is denoted by ‘r’. r is also a non-zero number. the general term or nth term of a geometric progression is an=arn1 

4. How can we form Arithmetico - Geometric series?

We can form Arithmetico - Geometric series by multiplying the corresponding element of arithmetic and geometric series.

5. How do you determine the first term of an infinite arithmetic geometric series?
The first term of an infinite arithmetic geometric series is the product of the first terms of its constituent arithmetic and geometric sequences. It's often denoted as a₁r, where a₁ is the first term of the arithmetic sequence and r is the first term of the geometric sequence.
6. How does an infinite arithmetic geometric series differ from a regular infinite geometric series?
An infinite arithmetic geometric series combines both arithmetic and geometric progressions, while a regular infinite geometric series only involves a geometric progression. This combination can lead to more complex behavior and different convergence conditions.
7. How does changing the first term of the arithmetic sequence affect the sum of an infinite arithmetic geometric series?
Changing the first term of the arithmetic sequence affects the starting point and scale of the series. It can significantly impact the sum if the series converges, as it influences every term in the series.
8. Is it possible for an infinite arithmetic geometric series to oscillate without converging?
Yes, an infinite arithmetic geometric series can oscillate without converging if the geometric part alternates in sign (r < -1) and the arithmetic part doesn't dominate. This results in terms that alternate in sign without diminishing sufficiently in magnitude.
9. How do you interpret the partial sums of an infinite arithmetic geometric series?
Partial sums represent the sum of a finite number of terms in the series. As more terms are added, the partial sums may approach the limit of the infinite series (if it converges) or may grow without bound (if it diverges).
10. What is an infinite arithmetic geometric series?
An infinite arithmetic geometric series is a combination of arithmetic and geometric sequences that continues indefinitely. It's formed by multiplying each term of an arithmetic sequence by the corresponding term of a geometric sequence.
11. How does the formula for the sum of an infinite arithmetic geometric series differ from finite series?
The formula for an infinite series includes a limit as the number of terms approaches infinity, while a finite series has a fixed number of terms. The infinite series converges only if certain conditions are met, unlike finite series which always have a definite sum.
12. What conditions must be met for an infinite arithmetic geometric series to converge?
For an infinite arithmetic geometric series to converge, the absolute value of the common ratio of the geometric part must be less than 1 (|r| < 1). Additionally, the arithmetic part should not grow faster than the geometric part decreases.
13. Can you explain the concept of convergence in the context of infinite arithmetic geometric series?
Convergence in an infinite arithmetic geometric series means that as more terms are added, the sum approaches a finite limit. This occurs when the terms become progressively smaller, eventually becoming negligible in their contribution to the total sum.
14. How does the common difference of the arithmetic sequence affect the sum of an infinite arithmetic geometric series?
The common difference of the arithmetic sequence influences the rate at which the terms in the series change. A larger common difference can lead to faster growth in the arithmetic part, potentially affecting convergence or the final sum if the series converges.
15. What is the relationship between the rate of convergence and the common ratio in an infinite arithmetic geometric series?
The rate of convergence is closely tied to the common ratio. A smaller absolute value of the common ratio (closer to 0) generally leads to faster convergence, as the terms decrease more rapidly.
16. How does the behavior of an infinite arithmetic geometric series change if the signs of terms alternate?
Alternating signs can affect the convergence and sum of the series. It may lead to oscillation or, in some cases, convergence to a different value than if all terms were positive. The behavior depends on the relative strengths of the arithmetic and geometric parts.
17. What is the importance of the ratio test in analyzing infinite arithmetic geometric series?
The ratio test is crucial for determining the convergence of the series. It involves examining the limit of the ratio of consecutive terms as n approaches infinity. If this limit is less than 1 in absolute value, the series converges.
18. Can you explain the concept of telescoping in the context of infinite arithmetic geometric series?
Telescoping in infinite arithmetic geometric series refers to the cancellation of terms when partial sums are calculated. This technique can sometimes simplify the analysis of the series, especially when dealing with differences between consecutive terms.
19. How does the concept of a generating function relate to infinite arithmetic geometric series?
Generating functions can be powerful tools for analyzing infinite arithmetic geometric series. They provide a way to represent the entire series as a single function, which can be manipulated to find sums, study convergence, or derive other properties of the series.
20. How does the concept of a limit superior (limsup) apply to infinite arithmetic geometric series?
The limit superior (limsup) of the sequence of terms in an infinite arithmetic geometric series can provide information about the series' convergence. If the limsup is less than 1, it guarantees absolute convergence of the series.
21. What role does the concept of dominated convergence play in understanding infinite arithmetic geometric series?
Dominated convergence in the context of infinite arithmetic geometric series refers to situations where the series is bounded by another convergent series. This concept can be useful in proving convergence, especially when direct computation of the sum is difficult.
22. What is the significance of the radius of convergence in relation to infinite arithmetic geometric series?
The radius of convergence, typically associated with power series, can be applied to certain forms of infinite arithmetic geometric series. It defines the range of values for which the series converges, providing insights into the series' behavior for different parameters.
23. Can you explain the concept of uniform convergence in the context of infinite arithmetic geometric series?
Uniform convergence in infinite arithmetic geometric series refers to the convergence of the series being consistent across a range of parameters. This concept is important when dealing with series that depend on variables, ensuring that the convergence behavior is well-behaved throughout a specified domain.
24. What is the relationship between infinite arithmetic geometric series and differential equations?
Infinite arithmetic geometric series can arise as solutions to certain differential equations, particularly those with constant coefficients. Understanding these series helps in solving and interpreting solutions to such differential equations.
25. What is the significance of the Cauchy product in relation to infinite arithmetic geometric series?
The Cauchy product is relevant when multiplying two infinite series, which can include arithmetic geometric series. It provides a method for determining the product series and can be useful in analyzing more complex series derived from arithmetic geometric series.
26. How does the concept of analytic continuation apply to infinite arithmetic geometric series?
Analytic continuation allows the extension of the domain of convergence for certain infinite arithmetic geometric series. This technique can provide meaningful interpretations of the series beyond its original radius of convergence, offering insights into its behavior in complex domains.
27. How do you analyze the behavior of an infinite arithmetic geometric series near its point of divergence?
Analyzing behavior near the point of divergence involves examining how the series terms grow or oscillate as they approach the critical values where convergence fails. This often requires careful limit analysis and can reveal interesting transitional behaviors.
28. Can you explain the concept of Abel summation in the context of infinite arithmetic geometric series?
Abel summation is a method for assigning sums to certain divergent series, including some arithmetic geometric series. It involves introducing a convergence factor and taking a limit, potentially giving meaning to series that don't converge in the traditional sense.
29. How does the study of infinite arithmetic geometric series relate to complex analysis and contour integration?
In complex analysis, infinite arithmetic geometric series can be studied using contour integration techniques. This approach can provide powerful methods for summing series, analyzing convergence in the complex plane, and connecting these series to other areas of mathematics.
30. What role does the common ratio of the geometric sequence play in an infinite arithmetic geometric series?
The common ratio of the geometric sequence is crucial in determining whether the series converges. If |r| < 1, it ensures that the geometric part of each term decreases rapidly enough for the series to potentially converge.
31. What is the significance of the limit in the formula for the sum of an infinite arithmetic geometric series?
The limit in the formula represents the behavior of the series as the number of terms approaches infinity. It helps determine whether the series converges to a finite sum or diverges to infinity.
32. Can an infinite arithmetic geometric series have a negative sum?
Yes, an infinite arithmetic geometric series can have a negative sum if the series converges and the combination of terms results in a negative value. This depends on the initial terms and the rates of change in both the arithmetic and geometric parts.
33. What happens to the sum of an infinite arithmetic geometric series if the common ratio is exactly 1?
If the common ratio is exactly 1, the geometric part becomes constant, and the series behaves like an arithmetic series. In this case, the series will diverge unless the arithmetic part is zero.
34. Can an infinite arithmetic geometric series converge to zero?
Yes, an infinite arithmetic geometric series can converge to zero if the terms become progressively smaller and their sum approaches zero as more terms are added. This requires specific relationships between the arithmetic and geometric parts.
35. How do you determine if an infinite arithmetic geometric series is absolutely convergent?
To determine absolute convergence, consider the series of absolute values of the terms. If this series converges, the original series is absolutely convergent. This often involves analyzing the geometric part independently of the arithmetic part.
36. What role does the concept of limits play in understanding infinite arithmetic geometric series?
Limits are fundamental in analyzing infinite series. They help determine whether the series converges to a finite value, diverges to infinity, or oscillates. The limit of partial sums as n approaches infinity defines the sum of the series if it exists.
37. How does the sum formula for an infinite arithmetic geometric series relate to the sum formulas for arithmetic and geometric series separately?
The sum formula for an infinite arithmetic geometric series combines elements from both arithmetic and geometric series formulas. It typically involves a fraction where the numerator includes terms from the arithmetic progression, and the denominator includes terms from the geometric progression.
38. How does the behavior of an infinite arithmetic geometric series change as the common ratio approaches 1?
As the common ratio approaches 1, the geometric part of the series becomes less influential in causing convergence. The series may transition from converging to diverging, with the arithmetic part playing a more dominant role in the series' behavior.
39. What is the significance of the arithmetic-geometric mean in relation to infinite arithmetic geometric series?
The arithmetic-geometric mean can sometimes appear in the analysis of infinite arithmetic geometric series, especially when looking at the relationship between consecutive terms. It can provide insights into the series' behavior and convergence properties.
40. How do you determine the rate of growth or decay in an infinite arithmetic geometric series?
The rate of growth or decay is determined by comparing the rates of change in the arithmetic and geometric parts. If the geometric decay (|r| < 1) outpaces the arithmetic growth, the series may converge. Otherwise, it may diverge or exhibit more complex behavior.
41. Can an infinite arithmetic geometric series have complex terms?
Yes, an infinite arithmetic geometric series can have complex terms. This often occurs when dealing with applications in physics or engineering. The analysis of convergence becomes more complex, involving the absolute value of complex numbers.
42. What is the effect of scaling all terms in an infinite arithmetic geometric series by a constant?
Scaling all terms by a constant factor multiplies the sum of the series (if it converges) by that constant. It doesn't change the convergence properties of the series but affects the magnitude of the sum.
43. How do you find the sum of an infinite arithmetic geometric series where the arithmetic sequence starts with a non-zero term?
To find the sum when the arithmetic sequence doesn't start at zero, you typically need to adjust the standard formula. This often involves incorporating the initial term of the arithmetic sequence into the numerator of the sum formula.
44. What is the relationship between infinite arithmetic geometric series and power series?
Infinite arithmetic geometric series can be viewed as a special case of power series where the coefficients form an arithmetic sequence. This connection allows techniques from power series analysis to be applied to certain arithmetic geometric series.
45. How does the concept of absolute convergence apply to infinite arithmetic geometric series?
Absolute convergence in an infinite arithmetic geometric series means that the series of absolute values of the terms converges. This is a stronger condition than regular convergence and implies that the series will converge regardless of the order of its terms.
46. Can you explain how to derive the sum formula for an infinite arithmetic geometric series?
Deriving the sum formula involves writing out the general term of the series, creating a telescoping series by subtracting consecutive partial sums, and then taking the limit as the number of terms approaches infinity. This process combines techniques from both arithmetic and geometric series.
47. What is the significance of the ratio of consecutive terms in an infinite arithmetic geometric series?
The ratio of consecutive terms provides insight into the series' behavior. If this ratio approaches a value less than 1 in absolute value as n increases, it suggests the series may converge. The ratio combines both the arithmetic and geometric aspects of the series.
48. How do you determine if an infinite arithmetic geometric series is conditionally convergent?
A series is conditionally convergent if it converges but does not converge absolutely. For an arithmetic geometric series, this occurs when the series of absolute values diverges, but the original series converges due to cancellations between positive and negative terms.
49. How does the behavior of an infinite arithmetic geometric series change if you alternate the signs of either the arithmetic or geometric progression?
Alternating signs in either progression can significantly affect the series' behavior. It may change convergence properties, alter the sum (if convergent), or introduce oscillatory behavior. The specific effect depends on how the alternation interacts with the existing progressions.
50. How do you analyze the convergence of an infinite arithmetic geometric series with variable terms?
Analyzing convergence with variable terms involves examining how the variables affect both the arithmetic and geometric parts. Techniques like the ratio test or root test are often applied, considering the limiting behavior of the terms as influenced by the variables.
51. How does the concept of a generating function apply to infinite arithmetic geometric series?
Generating functions provide a powerful tool for analyzing infinite arithmetic geometric series. By representing the series as a function, typically involving powers of a variable, properties like sums and convergence can be studied through function analysis techniques.
52. Can an infinite arithmetic geometric series have a periodic sum sequence?
Yes, an infinite arithmetic geometric series can have a periodic sum sequence under certain conditions. This typically occurs when the geometric part introduces periodicity that interacts with the arithmetic progression in a repeating pattern.
53. How do you determine the rate of convergence for an infinite arithmetic geometric series?
The rate of convergence is typically determined by analyzing how quickly the terms approach zero. This involves examining the ratio of consecutive terms or using asymptotic analysis to understand the behavior of the partial sums as the number of terms increases.
54. What role do infinite arithmetic geometric series play in approximation theory?
Infinite arithmetic geometric series are used in approximation theory to represent or approximate more complex functions. They can provide tractable series expansions for functions, allowing for easier analysis or computation in various mathematical and scientific applications.

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