The numbers of letters mailed forms a $G.P.:$ $4,4^{2}, \ldots .4^{8}$

First term $=4$

Common ratio $=4$

Number of terms $=8$

It is known that the sum of n terms of a $G.P.$ is given by

$S_{n}=\frac{a\left(r^{n}-1\right)}{r-1}$

$\therefore S_{8}=\frac{4\left(4^{8}-1\right)}{4-1}$

$=\frac{4(65536-1)}{3}=\frac{4(65535)}{3}=4(21845)=87380$

It is given that the cost to mail one letter is $50$ paisa.

$\therefore $ Cost of mailing $87380$ letters $= Rs .87380 \times \frac{50}{100}= Rs .43690$

Thus, the amount spent when $8^{\text {th }}$ set of letter is mailed is $Rs.$ $43690$ .