Arithmetico-Geometric Progression

Scoring high in the IIT JEE Main exam is a dream of many, which requires dedicated efforts and continuous practice to shine like a star. To help you in your preparation journey and make the ride smoother for you, Unacademy has come up with a range of chapters and topics which will help you understand each concept in detail as a thorough understanding is a must to ace the IIT JEE Mains exam .

If you’re someone who has a good hand in math, then you might have understood the Arithmetico-Geometric progression. However, there is an equal proportion of students who are scared of math as well as the Arithmetico-Geometric progression. When it comes to national-level examinations, every mark matters. Consequently, there’s no way you can leave any topic. To help you prepare for the Arithmetico-Geometric progression, Unacademy has come up with easy-to-understand solutions which will empower you while appearing for the JEE Mains. 

Arithmetico-Geometric Progression

In Mathematics, Arithmetico-Geometric progression can be defined as the result of geometric progression term-by-term multiplication with the terms corresponding to the arithmetic progression. It can be said that arithmetico–geometric progression’s nth term is the product of the nth term of both geometric and arithmetic progression. There are several applications in which the Arithmetico–geometric progression arises, including in probability theory while computing the expected values. The sequence is- 

Here, it can be seen that the numerator is highlighted with blue; however, the denominator is green. The summation of the following infinite sequence is referred to as an arithmetico–geometric series, which is also termed as Gabriel’s staircase.

Objects which showcase the characteristics of both geometric and arithmetic progression, the domination is applied to it. One of the common examples is which generalizes both geometric and arithmetic progression. These are also the linear difference equations in exceptional cases. 

Terms of the Sequence 

The arithmetico progression’s first few terms are composed of an AP or arithmetic progression highlighted in blue, with n as the initial value and d as the difference. However, the geometric progression is highlighted with green which r as the common ratio and b as the initial value. 

Sum of the terms arithmetico geometric progression 

The arithmetico-geometric progression first n terms sum has the form- 

Proof by Multiplying 

Arithmetico Geometric Series

An arithmetico geometric series can be obtained by the multiplication of the term-by-term of a GP along with the corresponding terms of an Arithmetico progression. Below, we have mentioned the general term of an arithmetico geometric. Let’s have a look- 

Sn = a + [a + d]r + [a + 2d]r2 + [a + 3d]r3 + [a + 4d]r4 + ………. + [a + (n – 1)d]rn-1 

1 + 3x + 6×2 + 9×3 + 12×4 +15×5 + 18×6 

The above given sequence is an arithmetico geometric sequence. Here, the arithmetico progression sequence is 1 + 3 + 6 + 9 + 12 + 15 + 18 whereas the geometric progression sequence is 1 + x + x2 + x3 + x4 + x5 + x6. 

Sum of an Infinite Arithmetico Geometric Series

Infinite Arithmetico Geometric sequence or series sum is . According to the Sum of an Infinite Arithmetico Geometric Series, d can be termed as the common difference of  whereas, the r is given as the common ratio of gn (|r| <1). Or, , where Sg is written as the infinite sum of the 9n

Difference Methods 

The image you’ve been given is a series p1, p2, p3, p4, …………pn  in a way that p2–p1, p2–p3, …….., pn –pn−1, either a geometric progression or an arithmetico progression. Also, the sum of this arithmetico geometric series need to be monitored through the below-mentioned steps- 

Find out the nth term of the given series 

Let, S =p1 + p2+ p3+ p4+ …………. + pn……… (1)

Also S = 0 + p1 + p2+ p3+ p4+ …………. + pn…….. (2) 

Equation (1) – Equation (2) we get

0 = p1 + (p2 –p1) + (p3 –p2) + (p3 –p4) + ………………. (pn−pn−1)–tn

Or tn = p1 + (p2 –p1) + (p3 –p2) + (p3 –p4) + ………………. (pn−pn−1)–tn

The results extracted are as follows- 

Sum of AGP

An arithmetic-geometric progression is defined as the progression where each and every term can be represented as the terms product of a geometric progression and arithmetic progression, also known as GP and AP, respectively. 

In order to find the sum of AGP, it can be calculated manually using some formulae. However, it can be a brainstorming task, especially when appearing for the IIT JEE Main exam, as terms can get extremely large or small at any point when you sum them up easily. Let’s understand the sum of AGP through a better and general approach through an example- 

Find the sum of the Arithmetico-Geometric progression sequence 1⋅2+2⋅22 +3⋅23 +⋯+100⋅2100.

If you’re thinking of solving these manually, the terms are too large for that process. 

Let the series sum by S then, S = 1⋅2+2⋅22 +3⋅23 +⋯+100⋅2100.

When S is multiplied by 2, 2S = 1⋅22 +2⋅23 +⋯+99⋅2100 +100⋅2101.

When 2S is subtracted from the S, the equation pops out to be 

Now, we have 

Conclusion 

Arithmetico-Geometric progression along with the Arithmetico Geometric Series, Infinite Arithmetico Geometric Series, and General term of AGP are some of the primary and most important concepts while studying for IIT JEE Mains as the exams have a vast syllabus which includes everything from class 9th to 12th and even more. 

Therefore, a thorough understanding of each and every concept is a must if you’re someone aiming to score good marks. Over the past ten years, it has been recorded that a decent amount of questions are being asked from this very chapter which means it definitely holds a lot of significance in every student’s life aiming to score higher grades.