Arithmetico-Geometric Progression

Definition: An Arithmetico-Geometric Progression (AGP) is a sequence in which each term is formed by multiplying the corresponding terms of an Arithmetic Progression (AP) and a Geometric Progression (GP).

Example: Consider an AP defined by the terms \(a, a+d, a+2d, \ldots, a+(n-1)d\) and a GP defined by the terms \(1, r, r^2, r^3, \ldots, r^{n-1}\). An AGP formed from these sequences will have terms that are the product of the corresponding terms from the AP and GP. Thus, the terms of the AGP will be \(a, (a+d)r, (a+2d)r^2, \ldots, [a+(n-1)d]r^{n-1}\).

For instance, if we have an AP with terms \(1, 3, 5, \ldots\) and a GP with terms \(1, x, x^2, \ldots\), then the AGP formed by these sequences will have terms \(1 \cdot 1, 3 \cdot x, 5 \cdot x^2, \ldots\) which simplifies to \(1, 3x, 5x^2, \ldots\).

The nth term (\(t_n\)) of an AGP can be obtained by multiplying the nth term of the AP with the nth term of the GP. Therefore, \(t_n = [a + (n-1)d] \cdot r^{n-1}\).

Consider the sequence \(10 \cdot 9^1, 9 \cdot 9^2, 8 \cdot 9^3, \ldots, 1 \cdot 9^{10}\). Each term is formed by multiplying the \(n\)th term of the AP \(10, 9, 8, \ldots, 1\) by the corresponding term of the GP \(9^1, 9^2, 9^3, \ldots, 9^{10}\). This sequence is an Arithmetico-Geometric Progression (AGP). The \(n\)th term of this AGP is given by \((11-n) \cdot 9^n\).

Finite Arithmetico Geometric Series

A finite Arithmetico-Geometric Series (AGS) is a series in which each term is the product of the corresponding terms of an Arithmetic Progression (AP) and a Geometric Progression (GP).

Consider the finite AGS up to \(n\) terms:

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

To find the sum \( S_n \), multiply both sides by \( r \):

\[ rS_n = ar + (a+d)r^2 + (a+2d)r^3 + \ldots + [a+(n-1)d]r^n \]

Subtract this new equation from the original series and isolate a geometric progression:

\[ \begin{align*} S_n - rS_n = &\left[ a + \color{red}(a+d)r + \color{blue}(a+2d)r^2 \color{black}+ \ldots + [a+(n-2)d]r^{n-2} + \color{green}[a+(n-1)d]r^{n-1} \right] \\ - &\left[ \qquad \color{red}ar + \color{blue}\qquad(a+d)r^2 \color{black}+ (a+2d)r^3 + \ldots + \qquad\qquad\color{green}[a+(n-2)d]r^{n-1} +\color{black} [a+(n-1)d]r^n \right] \end{align*} \]

When properly aligned, the subtraction yields:

\[ \begin{align*} (1-r)S_n &= a + dr + dr^2 + dr^3 + \ldots + dr^{n-2} + dr^{n-1} - [a+(n-1)d]r^n \\ &= a + dr \left( 1 + r + r^2 + \ldots + r^{n-2} \right) - [a+(n-1)d]r^n \end{align*} \]

Now, the sum in the parentheses is a geometric series with \(n-1\) terms, which can be summed using the formula for the sum of a geometric series:

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

This gives us the sum \( S_n \) for the finite AGS up to \(n\) terms.

Infinite AGP

An infinite Arithmetico-Geometric Progression (AGP) is a series where each term is the product of the corresponding terms of an infinite Arithmetic Progression (AP) and an infinite Geometric Progression (GP). The series does not terminate, and it is defined as:

\[ S = a + (a+d)r + (a+2d)r^2 + (a+3d)r^3 + \ldots \]

From the finite sum of an AGP, we have:

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

To find the sum of an infinite AGP, we take the limit as \( n \) approaches infinity. If \( |r| < 1 \), both \( r^{n-1} \) and \( r^n \) will approach zero as \( n \) becomes very large due to the properties of geometric progressions with a common ratio whose absolute value is less than 1. As a result, the terms containing \( r^{n-1} \) and \( r^n \) in the formula for \( S_n \) will vanish, leaving us with:

\[ S = \lim_{n \to \infty} S_n = \frac{a}{1-r} + \frac{dr}{(1-r)^2} \]

This is the sum of the infinite AGP, valid only if \( |r| < 1 \), ensuring convergence of the series.

Instead of remembering the above formula you can also use the following procedure, given that \(|r|<1\).

For an infinite Arithmetico-Geometric Progression (AGP) with common ratio \( |r| < 1 \), we can find the sum \( S \) by the method used previously:

Consider the infinite AGP:

\[ S = a + (a+d)r + (a+2d)r^2 + (a+3d)r^3 + \ldots \]

Multiply both sides by \( r \):

\[ rS = ar + (a+d)r^2 + (a+2d)r^3 + (a+3d)r^4 + \ldots \]

Now, subtract \( rS \) from \( S \):

\[ S - rS = a + (a+d)r + (a+2d)r^2 + \ldots \]

\[ - [ar + (a+d)r^2 + (a+2d)r^3 + \ldots ] \]

Aligning and subtracting term by term:

\[ (1 - r)S = a + dr + dr^2 + dr^3 + \ldots \]

The series on the right side is an infinite geometric series whose first term is \( dr \) and common ratio \( r \). The sum of this series is:

\[ \frac{dr}{1 - r} \]

Now, the equation becomes:

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

Solving for \( S \), we get:

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

This formula represents the sum of the infinite AGP for \( |r| < 1 \), ensuring the convergence of the series.

Example

Problem: Find the sum of the series \(1 + 2x + 3x^2 + 4x^3 + \ldots\) for \(|x| < 1\).

Solution:

Given the series:

\[ S = 1 + 2x + 3x^2 + 4x^3 + \ldots \]

This series is an infinite Arithmetico-Geometric Progression (AGP) with:

The arithmetic part having a common difference of \(d = 1\) (since the coefficients increase by 1 for each term).
The geometric part having a common ratio of \(r = x\).

From the formula for the sum of an infinite AGP:

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

For this series, \(a = 1\) (the first term of the AP), \(d = 1\) (the difference between consecutive terms of the AP), and \(r = x\) (the common ratio of the GP).

Substituting these values into the formula:

\[ S = \frac{1}{1 - x} + \frac{x}{(1 - x)^2} \]

Simplifying the expression:

\[ S = \frac{1}{(1 - x)^2} \]

Therefore, the sum of the series \(1 + 2x + 3x^2 + 4x^3 + \ldots\) for \(|x| < 1\) is \(\frac{1}{(1 - x)^2}\).