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A person writes a letter to four of his friends. He asks each one of them to copy the letter and mail to four different persons with instruction that they move the chain similarly. Assuming that the chain is not broken and that it costs $50$ paise to mail one letter. Find the amount spent on the postage when $8^{\text {th }}$ set of letter is mailed.
$Rs.$ $43690$
$Rs.$ $43690$
$Rs.$ $43690$
$Rs.$ $43690$
Solution
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 $Rs.$ $43690$ .
