Say No to Letters!

Algebra Level 3

A person writes a letter to each of his four friends. He asks each one of them to copy the letter and mail to four different persons with the instructions that they move the chain similarly. Assuming that the chain is not broken and that it costs 50 cents to main one letter. Find the total amount spent on the postage till the 8 th 8^{\text{th}} set of letters is mailed.

Details and Assumptions :

  • $ 1 = 100 \$ 1 = 100 cents.

  • Submit your answer in $ \$ .


The answer is 43690.

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Samara Simha Reddy by Samara Simha Reddy , 22, India

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1 solution

The number of letters mailed forms a GP.

The sequence if of the form 4 , 4 2 , . . . . . . 4 8 . 4, \, 4^2, \, ...... \, 4^8.

a = 4 , d = 4 , n = 8 a = 4, \, d = 4, \, n = 8

Sum of n terms in G.P., S n = a ( r n − 1 ) r − 1 ∴ S 8 = 4 ( 4 8 − 1 ) 4 − 1 = 4 ( 65536 − 1 ) 3 = 4 × 65535 3 ⟹ S 8 = 87380 \large \displaystyle \text{Sum of }n \text{ terms in G.P., } S_n = \frac{a \left(r^n - 1 \right)}{r-1}\\ \large \displaystyle \therefore S_8 = \frac{4 \left(4^8 - 1 \right)}{4-1} = \frac{4 \left(65536-1 \right)}{3} = \frac{4 \times 65535}{3}\\ \large \displaystyle \implies S_8 = \color{#3D99F6}{\boxed{87380}}

It is given that the cost to one mail is 50 50 cents.

∴ Cost of Mailing = $ 87380 × 50 100 = $ 43690 \large \displaystyle \therefore \text{Cost of Mailing} = \$ \, 87380 \times \frac{50}{100} = \color{#20A900}{\boxed{\$ \, 43690}}

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