What Is an Arithmetic Geometric Sequence Formula Derivation and Solved Examples
Sequence means an order. Now, the point is what is a sequence and how is it related to the subject of Mathematics. An arithmetic sequence is about numbers that add or subtract. The geometric progression goes from one term to another and multiplies or divides. The result is a shared value. Ref the figure below, in the arithmetic sequence the difference (d) is always a standard value 7. In the geometric series, the common value is always 2. An arithmetic sequence is about addition (or subtraction) in a set order. A geometric sequence is about multiplying (or division) in a set order.
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Arithmetic Sequence Formula
Let us understand the Arithmetic sequence formula. In the arithmetic sequence, one term goes to the next term by always adding—for example, 1, 2, 3, 4, 5 ….10, and so on. Here, in this sequence, each number moves to the second number by adding (or subtracting 1). Let us take some examples to understand better. 2, 5, 8, 11, 14….is arithmetic sequence as each step adds 3. The same holds for a reverse order (in subtraction). E.g. 7, 3, -1, -5 …is an arithmetic sequence as each step subtracts 4.
It is essential to note that the number that is added or subtracted at each level of an arithmetic sequence is called as the difference (d). The reason is that if you add or subtract (also known as finding the difference), you always get the same common value. Ref fig 2 below
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Geometric Sequence Formula
A geometric sequence or geometric series is a geometric order. We obtain results by multiplying the terms of a geometric sequence. In simple words, a geometric sequence moves from one term to the next by always multiplying (or by division) by the same common value or number. For example, 2, 4, 8, 16, 32 … is a geometric series. The reason being that each step multiplies by two. Let us take another example, 81, 27, 9, 3, 1 …it is a geometric sequence as each step divides (or multiplies) by the number 3.
It is essential to note that the common number that multiplies or divides at each step of a geometric sequence is called the ratio r. It is due to the fact that if you divide or find the ratio of the successive terms, you get a common or standard ratio. Ref Fig.3 below.
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Arithmetic Geometric Progression
An arithmetic progression is a series or sequence of numbers in which each term is derived from the next term, by adding or subtracting a fixed or common number called the common difference d. For instance, the series, 9, 6, 3, 0,-3 and so on, is an arithmetic progression with -3 as the standard difference. The progression -3, 0, 3, 6, 9 is an example of Arithmetic Progression (AP) that has three as the common difference d.
The established form of AP is a, a + d, a + 2d, a + 3d and ….so on. The nth term will be
an = a + (n -1)d
A geometric progression is a series in which each term is obtained by a multiplication or division of the next term, by a fixed or common number. For instance, the series 8, 4, -2, 1, -1/2 … is a Geometric Progression (GP) for which -1/2 is the common ratio.
The established form of GP is a, ar, ar2, ar3, ar4….and so on. The nth term will be an = ar (n-1)
Example -1
Work out the common difference and the next term of the following series:
3, 11, 19, 27, 35 …
We have to find the common difference d. you can pick up any pair. Let us start with subtractions.
11 - 3 = 8
19 -11 = 8
27 -19 = 8
35 - 27 = 8
Throughout, the difference is 8, so the common difference is 8.
We have 5 terms. We have to find the next or the 6th term. We can find out by adding the common difference to the fifth term, 35 + 8 = 43.
sixth term = 43 and common difference =8
Example -2
Find the common ratio and the 7th term of the following sequence
2/9, 2/3, 2, 6, 18.
We will take 6/2 = 3 and 18/6 = 3, the common ratio is 3, so r = 3
We have 5 terms, we have to find the 6th, then the 7th term.
So, a6 = 18 x 3 = 54
a7 = 54 x 3 = 162, the answer is common ratio r = 3 and seventh term = 162.
FAQs on Arithmetic Geometric Sequence Explained with Formula and Applications
1. What is an arithmetic geometric sequence?
An arithmetic geometric sequence (AGS) is a sequence formed by multiplying the terms of an arithmetic sequence and a geometric sequence together term by term. In general, its nth term is written as aₙ = (a + (n − 1)d)rⁿ⁻¹, where:
- a = first term of the arithmetic sequence
- d = common difference
- r = common ratio
- n = term number
It combines properties of both arithmetic progression (AP) and geometric progression (GP).
2. What is the formula for the nth term of an arithmetic geometric sequence?
The formula for the nth term of an arithmetic geometric sequence is aₙ = (a + (n − 1)d)rⁿ⁻¹. Here:
- a is the first arithmetic term
- d is the common difference
- r is the common ratio
This formula multiplies the nth term of an arithmetic sequence by the corresponding term of a geometric sequence.
3. How do you find the sum of an arithmetic geometric series?
The sum of an arithmetic geometric series is found using a special formula derived by multiplying the series by the common ratio and subtracting. For n terms, the sum is:
Sₙ = a(1 − rⁿ)/(1 − r) + dr(1 − rⁿ⁻¹)/(1 − r)², where r ≠ 1.
This formula applies when the terms follow (a + (n − 1)d)rⁿ⁻¹.
4. How is an arithmetic geometric sequence different from an arithmetic sequence?
An arithmetic sequence increases by adding a constant difference, while an arithmetic geometric sequence multiplies each arithmetic term by a geometric factor.
- Arithmetic sequence: aₙ = a + (n − 1)d
- Arithmetic geometric sequence: aₙ = (a + (n − 1)d)rⁿ⁻¹
The AGS grows faster or slower depending on the value of the common ratio r.
5. How is an arithmetic geometric sequence different from a geometric sequence?
A geometric sequence changes by multiplying by a constant ratio only, whereas an arithmetic geometric sequence includes both a changing arithmetic part and a geometric multiplier.
- Geometric sequence: aₙ = arⁿ⁻¹
- Arithmetic geometric sequence: aₙ = (a + (n − 1)d)rⁿ⁻¹
In AGS, the coefficient of rⁿ⁻¹ increases linearly.
6. Can you give an example of an arithmetic geometric sequence?
An example of an arithmetic geometric sequence is 2, 6, 12, 20, ...
Here:
- Arithmetic part: 2, 4, 6, 8, ... (a = 2, d = 2)
- Geometric part: 1, 2, 3, 4, ... does not apply directly, instead multiply arithmetic term by n
A clearer standard example using the formula is: if a = 1, d = 1, r = 2, then
- a₁ = (1)2⁰ = 1
- a₂ = (2)2¹ = 4
- a₃ = (3)2² = 12
- a₄ = (4)2³ = 32
So the sequence is 1, 4, 12, 32, ...
7. How do you derive the sum formula of an arithmetic geometric series?
The sum formula of an arithmetic geometric series is derived by multiplying the series by r and subtracting to eliminate terms.
- Let Sₙ = a + (a + d)r + (a + 2d)r² + ...
- Multiply by r and subtract from Sₙ
- Simplify the resulting expression
This algebraic method leads to the standard formula involving (1 − r) and (1 − r)² in the denominator.
8. What happens if the common ratio r equals 1 in an arithmetic geometric sequence?
If the common ratio r = 1, the arithmetic geometric sequence reduces to a simple arithmetic sequence.
- The nth term becomes aₙ = a + (n − 1)d
- The sum formula simplifies to the arithmetic series sum
- Sₙ = n/2 [2a + (n − 1)d]
So the geometric effect disappears when r equals 1.
9. When does an arithmetic geometric series converge?
An infinite arithmetic geometric series converges only if |r| < 1.
- If |r| < 1, the powers rⁿ approach 0
- If |r| ≥ 1, the terms do not approach zero
Convergence depends entirely on the magnitude of the common ratio r.
10. What are the applications of arithmetic geometric sequences?
Arithmetic geometric sequences are used in financial mathematics, compound interest models, and recurrence relations.
- Solving linear difference equations
- Calculating investment growth with increasing deposits
- Modelling processes with linear growth and exponential scaling
They commonly appear in advanced algebra, calculus, and mathematical modelling problems.
