Sum of an Infinite Arithmetic Geometric Series - Practice Questions & MCQ

The sum of an infinite AGP
$S_{\infty}$ denotes the sum of an infinite AGP. This sum is a finite quantity if $-1<r<1$

$
\mathrm{S}_{\infty}=a+(a+d) r+(a+2 d) r^2+(a+3 d) r^3 \ldots \ldots
$
Multiply both sides of eq (i) by 'r'

$
r \mathrm{~S}_{\infty}=a r+(a+d) r^2+(a+2 d) r^3+(a+3 d) r^4 \ldots \ldots
$
Subtract eq (ii) from eq (i)

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