• question_answer 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 instructions that they move the chain similarly. Assuming that the chain is not broken and that it costs $50\,\,paise$ to mail one letter. When the 8th set of letter is mailed, then the amount on postage will be A) $Rs.\,\,42780$                B) $Rs.\,\,43690$ C) $Rs.\,\,43680$                D)  None of these

Solution :

Firstly, one person post four letters to four of his friends. Then, those four persons post$4-4$ letters $i.e.,$ a total $4\times 4=16$ letters. Similarly, in next step $4\times 4\times 4=64$ letters are posted. Thus, there will be a $GP$ series. $i.e.,$ $4,\,\,16,\,\,64,\,\,256,\,....$ where,$a=4$and$r=4$ Now, total number of letters till 8th set                 ${{S}_{n}}=\frac{a({{r}^{n}}-1)}{r-1}$ $\Rightarrow$               ${{S}_{n}}=\frac{4({{4}^{8}}-1)}{4-1}$ $\Rightarrow$   ${{S}_{8}}=\frac{4}{3}({{4}^{8}}-1)$      $\left\{ \because \,\,{{S}_{n}}=\frac{\alpha ({{r}^{n}}-1)}{r-1},\,\,r>1 \right\}$ Cost of one letter$=Rs.\,\,0.50$ Hence, total cost$=\frac{4}{3}({{4}^{8}}-1)\times 0.50$                 $=\frac{4}{3}\times (65536-1)\times 0.50$                 $=\frac{4}{3}\times 65535\times 0.50$                 $=Rs.\,\,43690$

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