Question

A person writes a letter to four of his friends. He asked 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 the 8th set of the letter is mailed.

Answer 1

Hint: First, we have to find the total number of letters up to the 8th set. Since the cost of mailing one letter is 50 paise. From that we can determine the total cost of the letter mailing up to the 8th set by multiplying the total number of letters with the cost of one mail.

Complete step-by-step answer:
Now, we have to find the sequence of the letter posting.
The number of letters mailed in first set $ = 4$ letters
The number of letters mailed in second set $ = 4 \times 4 = 16$ letters
The number of letters mailed in third set $ = 16 \times 4 = 64$ letters
Thus, the sequence is $4,16,64, \ldots $
This sequence represents a G P as $\dfrac{{16}}{4} = 4\;{\rm{and}}\;\dfrac{{64}}{{16}} = 4$.
Let a represent the first term of the sequence and r be the common ratio.
Comparing the sequence 4, 16, 64 … we get $a = 4$ and $r = 4$.

Now, we have to find the sum of 8th set of letters for given a and r by using the formula ${S_n} = \dfrac{{a\left( {{r^n} - 1} \right)}}{{r - 1}}\;{\rm{for}}\;r > 1$

Substituting the value 4 for $a$, 4 for $r$ and 8 for$n$ in the above formula.
$\begin{array}{l}{S_n} = \dfrac{{a\left( {{r^n} - 1} \right)}}{{r - 1}}\\ \Rightarrow {S_8} = \dfrac{{4\left( {{4^8} - 1} \right)}}{{4 - 1}}\\ \Rightarrow {S_8} = \dfrac{{4\left( {65536 - 1} \right)}}{3}\\ \Rightarrow {S_8} = \dfrac{{4\left( {65535} \right)}}{3}\\ \Rightarrow {S_8} = 87380\end{array}$

Thus, the total number of letters mailed up to the 8th set is 87380.
Now, it is given that the mailing cost $ = 50$ paise per letter
The amount spent on mailing of 87380 letter $\begin{array}{c} = 50 \times 87380\\ = 4369000\end{array}$

Now, converting the 4369000 paise into rupees.
$\begin{array}{c}{\rm{Total}}\;{\rm{amount}} = \dfrac{{4369000}}{{100}}\\ = {\rm{Rs}}.43690\end{array}$
Hence, the amount spent on mailing 87380 letters is 43690 rupees.

Note: Geometric sequence, is a sequence of numbers where each term is equal to the multiplication of the previous number by a fixed, non-zero number called the common ratio. Here, we have to determine the total cost up to the 8th set of the mailing. Since total numbers of letters can be determined with the help of G.P. and the given cost per mail. Thus, it becomes easy to determine the total cost of mailing.