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प्रश्न
| A man 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. Assume that the chain is not broken and it costs ₹4 to mail one letter. |
1. The number of letters mailed in the 6th set of letters is:
(a) 2048
(b) 8192
(c) 4096
(d) 1024
2. The amount spent on the postage of 6th set of letters is:
(a) ₹16384
(b) ₹8192
(c) ₹32768
(d) ₹4096
3. The total number of letters mailed till the 5th set of letters is:
(a) 1024
(b) 4092
(c) 1236
(d) 1364
4. The amount spent on the postage till the 5th set of letters is:
(a) ₹16368
(b) ₹ 4096
(c) ₹5456
(d) ₹ 4944
5. The amount spent on the postage till the nth set of letters is:
(a) ₹ `[4/3 (4^n - 4)]`
(b) ₹ `[16/3 (4^n - 1)]`
(a) ₹ `[2/3 (4^n - 1)]`
(a) ₹ `[16 (4^n - 1)]`
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उत्तर
1. The correct option is (c)
Reasoning:
The number of letters mailed in consecutive sets forms a G.P.:
1st set: \[a = 4\]
2nd set: \[4 \times 4 = 4^2\]
Common ratio \[r = 4\]
The \[n^{\text{th}}\] term is given by: \[T_n = a r^{n - 1} = 4 \times 4^{n-1} = 4^n\]
For the 6th set \([n = 6]\): \[T_6 = 4^6 = 4096\]
2. The correct option is (a)
Reasoning:
Number of letters mailed in the 6th set = \[4096\]
Cost of mailing one letter = ₹4
Total amount spent on the 6th set: \[\text{Cost} = 4096 \times ₹4 = ₹16384\]
3. The correct option is (d)
Reasoning:
The total number of letters mailed till the 5th set is the sum of the first 5 terms of the G.P. where \[a = 4\], \[r = 4\] and \[n = 5\]:
\[S_n = \frac{a(r^n - 1)}{r - 1}\]
\[S_5 = \frac{4(4^5 - 1)}{4 - 1}\]
\[= \frac{4(1024 - 1)}{3}\]
\[= \frac{4 \times 1023}{3}\]
\[= 4 \times 341\]
\[= 1364\]
4. The correct option is (c)
Reasoning:
Total number of letters mailed till the 5th set = \[1364\]
Cost per letter = ₹4
Total amount spent: \[\text{Total Cost} = 1364 \times ₹4 = ₹5456\]
5. The correct option is (b)
Reasoning:
Total number of letters mailed till the \[n^{\text{th}}\] set is:
\[S_n = \frac{4(4^n - 1)}{4 - 1} = \frac{4}{3}(4^n - 1)\]
Since each letter costs ₹4, the total amount spent is:
\[\text{Amount} = 4 \times S_n = 4 \times \left[\frac{4}{3}(4^n - 1)\right] = ₹\left[\frac{16}{3}(4^n - 1)\right]\]
