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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 the 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 $8^{th}$ set of letter is mailed.

Toolbox:
• The $n^{th}$ term of a G.P. $=t_n=a.r^{n-1}$
We have to find the number of letters mailed in $8^{th}$ set.
Let us identify the sequence in which the letters are mailed.
No. of letters mailed in $1^{st}$ set $=4$
Given that each one has to mail for 4 different persons.
$\therefore$ No. of letters mailed in $2^{nd}$ set $=4\times 4=16$
Similarly no. of letters mailed in $3^{rd}$ set $=16\times 4=64$
and so on.
$\therefore$ The sequence in this process is $4,16,64,........$
This series is a G.P. with first term $a=4$ and common ratio $r=4$
$\Rightarrow\:$ The no. of letters mailed in $8^{th}$ set $=t_8$ of the above sequence.
We know that $n^{th}$ term of a G.P. $=t_n=a.r^{n-1}$
$i.e.,$ The no. of letters in the $8^{th}$ set $=4\times 4^{8-1}=4^8=63504$
Given the postage charge for each letter =50 paise= $Rs.\large\frac{1}{2}$
$\therefore$ The postage charges for $8^{th}$ set of letters $=63504\times \large\frac{1}{2}$