Arithmetic-Geometric Progression (AGP)
Arithmetic-Geometric Progression (AGP)
Arithmetic-Geometric Progression (AGP) is a sequence of terms that are generated by multiplying the corresponding terms of an arithmetic progression (AP) with a geometric progression (GP). This type of progression is less common than AP and GP but is an interesting and important concept in mathematics, especially in higher algebra and in solving certain types of problems.
Understanding AGP
An AGP is a sequence where each term is the product of the nth term of an AP and the nth term of a GP. To understand AGP, we first need to recall what AP and GP are:
- Arithmetic Progression (AP): A sequence in which each term after the first is obtained by adding a constant, called the common difference (d), to the previous term.
General form: $a, a+d, a+2d, a+3d, \ldots$
- Geometric Progression (GP): A sequence in which each term after the first is obtained by multiplying the previous term by a constant, called the common ratio (r).
General form: $a, ar, ar^2, ar^3, \ldots$
Now, let's define AGP:
- Arithmetic-Geometric Progression (AGP): A sequence in which each term is the product of the nth term of an AP and the nth term of a GP.
General form: $a, (a+d) \cdot r, (a+2d) \cdot r^2, (a+3d) \cdot r^3, \ldots$
Table of Differences and Important Points
| Feature | Arithmetic Progression (AP) | Geometric Progression (GP) | Arithmetic-Geometric Progression (AGP) |
|---|---|---|---|
| Definition | Sequence with a constant difference between consecutive terms. | Sequence with a constant ratio between consecutive terms. | Sequence with terms that are the product of corresponding terms of an AP and a GP. |
| General Term | $a_n = a + (n-1)d$ | $a_n = ar^{n-1}$ | $a_n = (a + (n-1)d) \cdot r^{n-1}$ |
| Common Difference/Ratio | Common difference (d) | Common ratio (r) | Both common difference (d) and common ratio (r) |
| Example | 2, 5, 8, 11, ... (d=3) | 3, 6, 12, 24, ... (r=2) | 2, 10, 24, 44, ... (d=3, r=2) |
Formulas in AGP
The nth term of an AGP can be found using the formula:
$$ a_n = (a + (n-1)d) \cdot r^{n-1} $$
where:
- $a$ is the first term of the AP,
- $d$ is the common difference of the AP,
- $r$ is the common ratio of the GP,
- $n$ is the term number.
The sum of the first n terms of an AGP is not as straightforward as in AP or GP. However, it can be calculated using certain techniques such as multiplying the AGP by the common ratio and subtracting it from the original sequence to create a telescoping series.
Examples to Explain Important Points
Example 1: Finding the nth Term
Given an AGP with the first term of the AP as 3, the common difference as 2, and the common ratio as ½, find the 5th term.
Using the formula for the nth term:
$$ a_5 = (3 + (5-1) \cdot 2) \cdot \left(\frac{1}{2}\right)^{5-1} $$
$$ a_5 = (3 + 8) \cdot \left(\frac{1}{2}\right)^4 $$
$$ a_5 = 11 \cdot \frac{1}{16} $$
$$ a_5 = \frac{11}{16} $$
Example 2: Sum of First n Terms
To find the sum of the first 4 terms of the AGP from Example 1, we can use the following approach:
Let $S$ be the sum of the first 4 terms. Then:
$$ S = 3 + (3+2) \cdot \frac{1}{2} + (3+4) \cdot \left(\frac{1}{2}\right)^2 + (3+6) \cdot \left(\frac{1}{2}\right)^3 $$
Now, multiply the entire series by the common ratio ($\frac{1}{2}$):
$$ \frac{1}{2}S = 3 \cdot \frac{1}{2} + (3+2) \cdot \left(\frac{1}{2}\right)^2 + (3+4) \cdot \left(\frac{1}{2}\right)^3 + (3+6) \cdot \left(\frac{1}{2}\right)^4 $$
Subtract the second equation from the first to get a telescoping series:
$$ \frac{1}{2}S = S - \left[3 + (3+2) \cdot \frac{1}{2} + (3+4) \cdot \left(\frac{1}{2}\right)^2 + (3+6) \cdot \left(\frac{1}{2}\right)^3\right] - (3+6) \cdot \left(\frac{1}{2}\right)^4 $$
Simplify and solve for $S$:
$$ \frac{1}{2}S = 3 - (3+6) \cdot \left(\frac{1}{2}\right)^4 $$
$$ S = 6 - \frac{9}{16} $$
$$ S = \frac{96}{16} - \frac{9}{16} $$
$$ S = \frac{87}{16} $$
Thus, the sum of the first 4 terms of the AGP is $\frac{87}{16}$.
AGP problems often require a combination of algebraic manipulation and knowledge of AP and GP properties. Practice with various examples is key to mastering AGP-related questions.