Question

Find the sum of infinite series $\left|\begin{array}{cc}1& 2\\ 6& 4\end{array}\right|+\left|\begin{array}{cc}1/2& 2\\ 2& 4\end{array}\right|+\left|\begin{array}{cc}1/4& 2\\ 2/3& 4\end{array}\right|+\dots \dots .$

Answer 1

Answer: -10.00

$=\left|\begin{array}{cc}1+\frac{1}{2}+\frac{1}{4}\dots \infty & 2\\ 6+2+\frac{2}{3}\dots \cdots \infty & 4\end{array}\right|$
Using a $+ar+a{r}^2\dots \infty =\frac{a}{1-r}$
$=\left|\begin{array}{cc}\frac{1}{1-\frac{1}{2}}& 2\\ \frac{6}{1-\frac{1}{3}}& 4\end{array}\right|=\left|\begin{array}{cc}2& 2\\ 9& 4\end{array}\right|=-10$