Question

# 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 the 8th set of letters is mailed.

Hint: In this question each person writes four letters to four different persons. And this process continues i.e., this process is in a sequence. Since all the common terms have the same ratio this progression is in geometric progression.

According to this question, one person writes a letter to his 4 friends.
Now each of these 4 friends will write letters to their 4 different friends. So, the number of letters written here will be ${4^2}$
And this process continues………
So, the number of letters written forms a geometric progression (G.P)
i.e., $4,{4^2},{4^3},......,{4^8}$
We know that if a series is in geometric progression (G.P) of ‘$n$’ terms, with first term ‘$a$’ and common ratio ‘$r$’ then the sum of the series is given by ${S_n} = \dfrac{{a\left( {{r^n} - 1} \right)}}{{r - 1}}$.
Since we have to find the cost of 8 set of letters, we have $a = 4,r = 4,n = 8$
So, ${S_n} = \dfrac{{4\left( {{4^8} - 1} \right)}}{{4 - 1}}$
${S_8} = \dfrac{{4\left( {65536 - 1} \right)}}{3} \\ {S_8} = \dfrac{{4\left( {65535} \right)}}{3} \\ {S_8} = \dfrac{{262140}}{3} \\ {S_8} = 87380 \\$
Given each mail costs 50 paise, then 87380 letters costs ${\text{Rs }}\dfrac{{50}}{{100}} \times 87380 = {\text{Rs }}43690$
Thus, the amount spent on postage when 8th set of letters mailed is Rs.43690

Note: In this problem we have converted 50 paise into rupees by the conversion 1 rupee equals to 100 paise. Always remember that the above used formula for the summation of the terms in geometric progression (G.P) is valid only when the common ratio is greater than one.