Step-by-Step Guide to Solving Arithmetic Geometric Sequences
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
1. What is an Arithmetic-Geometric Progression (AGP)?
An Arithmetic-Geometric Progression (AGP) is a special type of sequence in mathematics. Each term in an AGP is formed by multiplying the corresponding terms of an Arithmetic Progression (AP) and a Geometric Progression (GP). For instance, if you have an AP and a GP, the first term of the AGP is the product of the first terms of the AP and GP, the second term is the product of the second terms, and so on.
2. Can you provide a clear example of an Arithmetic-Geometric sequence?
Certainly. To form an Arithmetic-Geometric sequence, let's take a simple AP and GP.
- Consider the Arithmetic Progression (AP): 1, 3, 5, 7, ... (first term is 1, common difference is 2)
- Consider the Geometric Progression (GP): 2, 4, 8, 16, ... (first term is 2, common ratio is 2)
- 1st Term: 1 × 2 = 2
- 2nd Term: 3 × 4 = 12
- 3rd Term: 5 × 8 = 40
- 4th Term: 7 × 16 = 112
3. What is the general formula for the nth term of an Arithmetic-Geometric Progression?
The formula for the nth term of an AGP is a combination of the formulas for the nth term of an AP and a GP. If the Arithmetic Progression is given by a, a+d, a+2d, ... and the Geometric Progression is b, br, br², ..., then the nth term of the corresponding AGP is given by the formula:
Tn = [a + (n-1)d] × [brn-1]
Here, 'a' is the first term of the AP, 'd' is the common difference, 'b' is the first term of the GP, and 'r' is the common ratio.
4. How is the sum of an Arithmetic-Geometric Progression calculated?
Calculating the sum of an AGP involves a specific, clever method rather than a single direct formula. The steps are as follows:
- Let the sum of 'n' terms be Sn. Write out the full series.
- Multiply the entire equation for Sn by the common ratio 'r' of the geometric part. This gives you a new equation for rSn.
- Subtract the second equation from the first (i.e., calculate Sn - rSn).
- This subtraction cancels out the complex AGP structure and leaves a new series that is a simple GP.
- Calculate the sum of this new GP and solve for Sn to get the final answer.
5. What is the key difference that distinguishes an AGP from a simple AP or GP?
The key difference lies in the relationship between consecutive terms.
- In an AP, you add a constant value (common difference) to get the next term.
- In a GP, you multiply by a constant value (common ratio) to get the next term.
- In an AGP, there is no constant difference or constant ratio. The progression's behaviour is a hybrid; its terms are generated by both an additive and a multiplicative rule acting at the same time, making its growth pattern more complex than either an AP or a GP alone.
6. Under what conditions does the sum of an infinite Arithmetic-Geometric Progression have a finite value?
The sum of an infinite AGP converges to a finite value only if the absolute value of its common ratio 'r' (from the geometric part) is strictly less than one. The mathematical condition is |r| < 1. If the common ratio is 1 or greater, the terms of the series will not shrink towards zero, and the sum will grow infinitely large (diverge). This condition is crucial for finding the sum to infinity.
7. Why is the method of 'subtracting the series multiplied by the common ratio' so effective for finding the sum of an AGP?
This method is effective because it is designed to transform a complex problem into a simpler one. An AGP has two patterns (arithmetic and geometric) running at once. By multiplying the series by the common ratio 'r' and then subtracting, you strategically align the terms. This alignment cancels out the dual-pattern complexity, and the resulting series becomes a standard Geometric Progression. We can easily find the sum of a GP, which then allows us to solve for the original, more complex AGP sum.
8. What is the importance of studying Arithmetic-Geometric Progressions in mathematics?
While not as common as AP or GP in basic applications, AGP is an important concept in higher mathematics. Its importance includes:
- Problem-Solving Technique: It teaches the valuable technique of transforming complex series into simpler ones.
- Probability Theory: It is used to calculate the expected values in certain probability distributions where events have both increasing chances and decreasing weights.
- Financial Modelling: It can be applied to model certain types of complex financial instruments where periodic payments increase by a fixed amount but are also subject to a compounding discount factor.
