Arithmetico Geometric Series: Definition & Examples

Before learning the Arithmetic-Geometric Series, let's revise the concept of sequence. A sequence is formed when terms are written in order such that they follow a particular pattern. Understanding AGP involves understanding the different principles of AP and GP. In real life, AGP is applicable in areas such as population dynamics, algorithms, etc.

Arithmetic-Geometric Series

Arithmetico-geometric series is the combination of arithmetic and geometric series. This series is formed by taking the product of the corresponding elements of arithmetic and geometric series. In short form, it is written as A.G.P (Arithmetico-Geometric series).

Wherein-

An arithmetic series is a sequence in which each term increases or decreases by a constant term or fixed number. This fixed number is called the common difference of an AP and is generally denoted by ' $d$ '.

A geometric series or geometric sequence is a sequence where the first term is non-zero and the ratio between consecutive terms is always constant. The 'constant factor' is called the common ratio and is denoted by ' $r$ '. $r$ is also a non-zero number.

Properties of A.M. and G.M

A and G are arithmetic and geometric mean of ' $a$ ' and ' $b$ ', two real, positive and distinct numbers. Then,

• $a$ and $b$ are the roots of the equation $x^2-2 A x+G^2=0$.

• a and b are given by $A \pm \sqrt{(A+G)(A-G)}$.

Proof:

$
\begin{aligned}
& A=\frac{a+b}{2} \Rightarrow 2 A=a+b \\
& G=\sqrt{a b} \Rightarrow G^2=a b
\end{aligned}
$

$a$ and $b$ are the roots of the equation, then

$
\begin{aligned}
& x^2-2(\text { sum of roots }) x+\text { products of roots }=0 \\
& \Rightarrow x^2-(a+b)+a b=0 \\
& \Rightarrow x^2-2 A x+G^2=0
\end{aligned}
$
Roots of the equation are

$
\begin{aligned}
& x=\frac{2 A \pm \sqrt{(-2 A)^2-4 \cdot 1 \cdot G^2}}{2} \\
& x=A \pm \sqrt{(A+G)(A-G)}
\end{aligned}
$
Let the given AP be $a,(a+d),(a+2 d),(a+3 d)$, $\qquad$
And, the GP is $1, r, r^2, r^3$, $\qquad$
Multiplying the corresponding elements of the above series, we get, $a,(a+d) r,(a+2 d) r^2,(a+3 d) r^3, \ldots \ldots$

This is a standard Arithmetico-Geometric series.
Eg: $1,3 x, 5 x^2, 7 x^3, 9 x^4, \ldots \ldots$

The sum of n-terms of an Arithmetic-Geometric series

Let $S_n$ denote the sum of $n$ terms of a given sequence. Then,

$
\mathrm{S}_{\mathrm{n}}=a+(a+d) r+(a+2 d) r^2+\ldots \ldots+(a+(n-1) d) r^{n-1}
$
Multiply both side of eq (i) by ' r '

$
r \mathrm{~S}_{\mathrm{n}}=a r+(a+d) r^2+(a+2 d) r^3+\ldots+(a+(n-1) d) r^n
$
Subtract eq (ii) from eq (i)

$
\begin{aligned}
& (1-r) \mathrm{S}_{\mathrm{n}}=a+\left[d r+d r^2+d r^3+\ldots .+d r^{n-1}\right]-[a+(n-1) d] r^n \\
& \Rightarrow(1-r) \mathrm{S}_{\mathrm{n}}=a+d r\left(\frac{1-r^{n-1}}{1-r}\right)-[a+(n-1) d] r^n \\
& \Rightarrow \mathbf{S}_{\mathbf{n}}=\frac{\mathbf{a}}{\mathbf{1 - r}}+\mathbf{d r}\left(\frac{\mathbf{1}-\mathbf{r}^{\mathbf{n}-\mathbf{1}}}{(\mathbf{1}-\mathbf{r})^{\mathbf{2}}}\right)-\frac{[\mathbf{a}+(\mathbf{n}-\mathbf{1}) \mathbf{d}] \mathbf{r}^{\mathbf{n}}}{\mathbf{1}-\mathbf{r}}
\end{aligned}
$

The sum of infinite terms of an Arithmetic-Geometric series:

The infinite terms can not be solved mentally, so we will have to find a general approach.

Let's denote the AGP by: $a,(a+d) r,(a+2 d) r^2,(a+3 d) r^3, \ldots$
Here, $a$ is the first term of the arithmetic series, $d$ is the common difference of the arithmetic series, and $r$ is the common ratio of the geometric series.

To find the sum of the infinite AGP, we can use the following formula:

$
S=\frac{a}{1-r}+\frac{d r}{(1-r)^2}
$
This is the sum of an infinite arithmetic-geometric series.

Solved Example Based on Arithmetic-Geometric series

Example 1: $1+3+7+15+31+ . . . . .$ $\qquad$ to $n$ terms
1) $2^{n+1}-n$
2) $2^{n+1}-n-2$
3) $2^n-n-2$
4) None of these

Solution

$
\begin{aligned}
& S_n=1+3+7+15+31+\ldots \ldots+T_n \\
& S_n=\quad 1+3+7+\ldots \ldots \ldots \ldots+T_{n-1}+T_n
\end{aligned}
$
Subtracting

$
\begin{aligned}
& 0=1+2+4+8 \ldots(\text { nterms })-T_n \\
& T_n=1 \times \frac{\left(2^n-1\right)}{2-1} \\
& =\left(2^n-1\right) \\
& S_n=\sum_1^n T_n=\sum_1^n 2^n-\sum_1^n 1 \\
& =2\left(2^n-1\right)-n \\
& =2^{n+1}-(n+2)
\end{aligned}
$

Hence, the answer is the option (2).

Example 2: Example 2: $S$ is the sum of the first 9 terms
where $a \neq 0$ and $x \neq 1. \space {\text {If }} S=\frac{x^{10}-x+45 a(x-1)}{x-1}$ then k is

1) 3

2) 2

3) -3

4) -5

Solution

$S=(x+k a)+\left(x^2+(k+2) a\right)+\left(x^3+(k+4) a\right)+\left(x^4+(k+6) a\right)+\ldots$

$
=\left(x+x^2+x^3+---x^9\right)+a(k+(k+2)+(k+4)+\cdots(k+16))
$

First series is GP and second series becomes an AP after separating $k$ terms .So,
$
=\frac{x\left(x^9-1\right)}{(x-1)}+a[9 k+72]
$

Now, comparing this with $S=\frac{x^{10}-x+45 a(x-1)}{x-1}$

$\begin{aligned} & \therefore 9 k+72=45 \\ & k+8=5 \\ & k=-3\end{aligned}$

Hence, the answer is the option 3.

Example 3: Which of the following is not an AGP?

1) $1,2 x, 3 x^2, 4 x^3$
2) $1,1,3 / 4,1 / 2,5 / 16 \ldots$....
3) $3 a, 5 a^2, 7 a^3$
4) $1,3.2^2, 4.2^3, 5.2^4$

Solution

Options 1 and 3 are AGP

In Option 2, terms can be written as 1, 2/2, 3/4, 4/8, 5/16,... so it is an AGP

Option 4 is wrong, as for it to an AGP, the first term should have been 2.2 (=4)

Hence, the answer is the option (4).

Example 4: What is the sum of the first 10 terms of $1.2+2.2^2+3.2^3+\ldots \ldots$

1) $11.2^{11}+2$
2) $9.2^{11}+2$
3) $11.2^{11}-2$
4) $9.2^{11}-2$

Solution

$
S=1.2+2.2^2+3.2^3+\ldots . .10 .2^{10}
$
As the common ratio of the corresponding GP is 2 , so
$2 S=$

$
1.2^2+2.2^3+
$

$\qquad$ $10.2^{11}$

Subtracting these equations

$
\begin{aligned}
& -S=1\left(2+2^2+2^3+\ldots . .+2^{10}\right)-10.2^{11} \\
& S=10.2^{11}-\left(2+2^2+2^3+\ldots . .+2^{10}\right) \\
& S=10.2^{11}-2\left(2^{10}-1\right) \\
& =10.2^{11}-2^{11}+2=9.2^{11}+2
\end{aligned}
$

Hence, the answer is the option (2).

Summary

Arithmetico-geometric series is a fundamental concept in mathematics. By understanding the formulas and properties of AP and GP. It is used to generalize the pattern we see in day-to-day life. This sequence helps across every field, highlighting their significance in both theoretical and practical contexts.