Arithmetic Geometric Sequence: Formula & Sum of Terms

Arithmetic-geometric sequence is a sequence of progression in which each term can be represented as a product of the term of an arithmetic progression and a geometric progression. It is generally abbreviated as AGP. Now, if a₁,a₂,a₃,………..an be an arithmetic sequence and b₁, b₂, b₃,…………bn be a geometric progression, then a₁b₁, a₂b₂,a₃b₃,……… anbn is called as an arithmetic-geometric sequence and it is of the form;

a,(a+d)r,(a+2d)r²,(a+3d)r³,…,[a+(n−1)d]rⁿ⁻¹

where, a is the first term of the sequence, d is the common difference and r is the common ratio. The sum of the number of the finite and infinite Arithmetic Geometric series can also be computed.

Read More: Math Formulas

Key Takeaways- Arithmetic Sequence, Geometric Sequence, Arithmetic Geometric Sequence, Common Difference, Common Ratio

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What is Arithmetic Sequence?

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An arithmetic sequence can be regarded as an ordered set of numbers that have a common difference in terms of value between each consecutive term. Let us assume the sequence of 3,9,15,21,27… Here in this sequence, each number moves to the second number by adding or subtracting the value of 6. It is essential to mention here that the number that is added or subtracted at each level of the arithmetic sequence is called the difference and is usually represented by ‘d’.

Thus, an arithmetic sequence can be written in the following form:

a, a+d, a+2d,…………………….a +(n-2)d, a+(n-1)d

Here, 

a is the first term of the sequence

d is the common difference

a + (n-1) d is the nth term of Arithmetic Sequence.

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What is a Geometric Sequence?

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Geometric sequence or progression refers to a sequence of non-zero numbers where each term after the first one is found by multiplying the previous one by an unchangeable non-zero number, called a common ratio. Thus, in the case of a geometric sequence, each number moves from one term to the next by always multiplying or dividing by the same common value or number. For instance, 2,4,8,16, 32, 64…. is a geometric sequence as each number has to be multiplied by 2 in order to get the next number in the series. 

In mathematical terms, a geometric sequence can be written as;

a, ar, ar²………….ar^n-2, ar^n-1

Here, 

a is the first term of the sequence.

r is the common ratio

ar^n-1 is the nth term of the geometric sequence.

Read Also: Latus Rectum of Ellipse

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What is Arithmetic Geometric Sequence?

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An Arithmetic-geometric sequence is a sequence or progression in which each term can be represented as a product of the term of an arithmetic progression and a geometric progression.

Now, let a₁,a₂,a₃,………..a_n be an arithmetic sequence and b₁, b₂, b₃,…………b_n be a geometric progression, then a₁b₁, a₂b₂,a₃b₃,……… a_nb_n is called as an arithmetic geometric sequence and it is of the form;

a, (a+d)r, (a+2d)r², (a+3d)r³,…,[a+(n−1)d]rⁿ⁻¹

Here, a is the first term of the sequence, d is the common difference and r is the common ratio.

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Sum of Terms of Arithmetic Geometric Sequence

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The sum of the first n terms of an arithmetic geometric sequence can be written in the following form-

Here, S_n is the sum of the terms of the sequence and A_n and G_n are the nth term of the arithmetic and the geometric sequence respectively.

Read Also: Permutations and Combinations

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Proof

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Multiplying the sequence 

S_n= a+ | a+d |r + |a + 2d| r² + …….+ [a +(n-1)d]r^n-1

by ‘r’ we get,

Now, subtracting rS_n from S_n we get,

rS_n= ar + |a + 2d | r³ +…...+[a+(n-1)d]rⁿ

Here, the last equality results from the expression of the sum of geometric progression. Dividing the expression by (1-r) gives the result.

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Sum of AGP up to Infinite Terms

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So far, we have found the sum of finitely many terms. Now let us consider an infinite many terms. As we cannot manually sum up an infinite number of terms, we will put a general approach here.

If -1 < r < 1, then the sum S of the arithmetic geometric series of infinitely many terms can be given by; 

In other terms, the sum of an AGP is generally given by,

S(1 – r) = a – [a + (n – 1)d] rⁿ + \(\frac{dr(1-r^{n-1})}{1-r}\)

S = \(\frac{a-[a+(n-1)d]r^n}{(1-r)}\) + \(\frac{dr(1-r^{n-1})}{(1-r)^2}\)

Now it has to be mentioned here that if r 1. Then the term a+n-1drn-1 gets very large and the sum does not converge. So here, we have to consider r < 1 and we will be using;

 rⁿ = 0

As we have already discussed earlier, the sum of the first n terms in an AGP can be represented by;

S_n = \(\frac{a-[a+(n-1)d]r^n}{(1-r)}\) + \(\frac{dr(1-r^{n-1})}{(1-r)^2}\)

Using it we get,

S_n = \(\frac{a}{1-r} + \frac{dr(1-0)}{(1-r)^2}\)

S_n = \(\frac{a}{1-r} + \frac{dr(1-0)}{(1-r)^2}\) (where, r < 1)

Read Also: Class 10 Mathematics Chapter 5 Arithmetic Progressions

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Things to Remember

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• An Arithmetic Sequence can be regarded as an ordered set of numbers that have a common difference in terms of value between each consecutive term. 
• An arithmetic sequence can be written in the following form:

a, a+d, a+2d,…………………….a +(n-2)d, a+(n-1)d

• A Geometric Sequence refers to a sequence of non-zero numbers where each term after the first one is found by multiplying the previous one by an unchangeable non-zero number, called a common difference.
• Mathematically, a geometric sequence can be written as;

a, ar, ar²………….ar^n-2, ar^n-1

• An Arithmetic- geometric sequence is a sequence of progression in which each term can be represented as a product of the term of an arithmetic progression and a geometric progression.
• An Arithmetic- Geometric sequence can be written as;

a,(a+d)r,(a+2d)r²,(a+3d)r³,…,[a+(n−1)d]rⁿ⁻¹

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