Question

Find the sum of infinite series $\begin{vmatrix}1 & 2 \\ 6 & 4\end{vmatrix}+\begin{vmatrix}1 / 2 & 2 \\ 2 & 4\end{vmatrix}+\begin{vmatrix}1 / 4 & 2 \\ 2 / 3 & 4\end{vmatrix}+\ldots \ldots .$

Answer 1

$=\begin{vmatrix}1+\frac{1}{2}+\frac{1}{4} \ldots \infty & 2 \\ 6+2+\frac{2}{3} \ldots \cdots \infty & 4\end{vmatrix}$
Using a $+ar+ ar ^{2} \ldots \infty=\frac{ a }{1- r }$
$=\begin{vmatrix}\frac{1}{1-\frac{1}{2}} & 2 \\ \frac{6}{1-\frac{1}{3}} & 4\end{vmatrix}=\begin{vmatrix}2 & 2 \\ 9 & 4\end{vmatrix}=-10$