Arithmetico Geometric Series - Practice Questions & MCQ

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

Let the given AP be $a,(a+d),(a+2 d),(a+3 d)$,
And, the GP is $1, r, r^2, r^3, \ldots \ldots$
Multiplying the corresponding elements of the above progression, we get, $a,(a+d) r,(a+2 d) r^2,(a+3 d) r^3, \ldots \ldots$
This is a standard Arithmetico-Geometric Progression.
Eg: $1,3 x, 5 x^2, 7 x^3, 9 x^4$,
The sum of n-terms of an Arithmetic-Geometric Progression
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 sides 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}-\mathbf{r}}+\mathbf{d r}\left(\frac{1-\mathbf{r}^{\mathbf{n}-\mathbf{1}}}{(1-\mathbf{r})^2}\right)-\frac{[\mathbf{a}+(\mathbf{n}-\mathbf{1}) \mathbf{d}] \mathbf{r}^{\mathbf{n}}}{\mathbf{1}-\mathbf{r}}
\end{aligned}
$