Arithmetic Geometric Progression


We learnt about Arithmetic Progression , and Geometric Progression  . In this chapter we will learn about a new type of series which is a mix of both, called Arithmetic-Geometric Progressions.

I. Introduction
In this series each term is a product of the corresponding terms of an Arithmetic Progression and a Geometric Progression.

Let us say we have an Arithmetic Progression $A$, defined by the first term $a$ and the common difference $d$, whose $n \xasuper{th}$ term is obtained by $A_n = a + (n - 1)d$, and a Geometric Progression $G$, defined by the first term $A$ and a common ratio $r$, whose $n \xasuper{th}$ term is obtained by $G_n = A r^{n - 1}$

The combined Arithmetic-Geometric progression, $C$ can be defined such that the $n \xasuper{th}$ term is obtained as $C_n = A_n \times G_n$ for each $n$. 

Therefore, we get the series as
$C_n = \{a + (n - 1)d\}(A r^{n - 1})$

$\Rightarrow C_n = aA r^{n - 1} + (n-1)dA r^{n - 1}$

$\Rightarrow C_n = aA r^{n - 1} - dA r^{n - 1} + ndA r^{n - 1}$

$\Rightarrow C_n = (aA - dA) r^{n - 1} + dAn r^{n - 1}$

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Since $a, A, d$ are constants, we can substitute $aA - dA = k_1$ and $dA = k_2$, and get:
$C_n = k_1 r^{n-1} + k_2n r^{n-1}$


II. Summation Of Finite AG Progression
Now that we know the definition of an AG progression, we will see how to derive the formula for the summation of the first $n$ terms of the series.
$\sum \limits_{i = 1}^{n}\left(k_1 r^{i-1} + k_2i r^{i-1}\right)$

$= \sum \limits_{i = 1}^{n}k_1 r^{i-1} + \sum \limits_{i = 1}^{n}k_2i r^{i-1}$

The first part of the above sum is a simple GP series, and we already know that the sum will be $S_1 = \dfrac{k_1(r^n - 1)}{r - 1}$, so let us focus on the second part of the series which is:
$\sum \limits_{i = 1}^{n}k_2i r^{i-1}$

Let
$S_2 = \sum \limits_{i = 1}^{n}k_2i r^{i-1} = k_2(1\cdot r^0 + 2\cdot r^1 + 3\cdot r^2... + n\cdot r^{n - 1})$

$\therefore r \times S_2 = k_2(1\cdot r^1 + 2\cdot r^2 + 3\cdot r^3... + n\cdot r^{n})$ 

$\therefore S_2 - r \times S_2= k_2(1\cdot r^0 + 1\cdot r^1 + 1\cdot r^2... + 1\cdot r^{n - 1} - n\cdot r^{n})$

$\Rightarrow S_2(1 - r) = k_2 \left(r^0 + r^1 + r^2... r^{n - 1} - n\cdot r^{n}\right)$

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$\Rightarrow S_2(1 - r) = k_2 \left(\dfrac{r^n - 1}{r - 1} - n\cdot r^{n}\right)$

$\Rightarrow S_2(1 - r) = k_2 \left(\dfrac{r^n - 1 - nr^{n + 1}  + nr^n}{r - 1}\right)$

$\Rightarrow S_2(r-1) = k_2 \left(\dfrac{-r^n + 1 + nr^{n + 1}  - nr^n}{r - 1}\right)$

$\Rightarrow S_2 = k_2 \left(\dfrac{nr^n(r - 1) - (r^n - 1)}{(r - 1)^2}\right)$

Now we can combine the two parts $S_1$ and $S_2$ to get the overall sum as: 

$\sum \limits_{i = 1}^{n}\left(k_1 r^{i-1} + k_2i r^{i-1}\right) = S_1 + S_2$

$= \dfrac{k_1(r^n - 1)}{r - 1} + k_2 \left(\dfrac{nr^n(r - 1) - (r^n - 1)}{(r - 1)^2}\right)$


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III. Summation Of Infinite AG Progression
Like GP series, if $|r| \lt 1$ then the series converges to a finite value as $n$ grows large, and the sum of such a series also converges to a finite value.
Since $|r| < 1$, $r^n$ approaches $0$ as $n$ grows large.
Substituting $r^n = 0$ in the expression for the sum of finite series in the previous section we get:
$\sum \limits_{i = 1}^{\infty}\left(k_1 r^{i-1} + k_2i r^{i-1}\right)$

$= \dfrac{k_1(0 - 1)}{r - 1} + k_2 \left(\dfrac{n\cdot 0 \cdot (r - 1) - (0 - 1)}{(r - 1)^2}\right)$

$= \dfrac{k_1( - 1)}{r - 1} + k_2 \left(\dfrac{1}{(r - 1)^2}\right)$

$= \dfrac{k_1}{1 - r} + \dfrac{k_2}{(1 - r)^2}$,

where $k_1 = aA - dA$ and $k_2 = dA$