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 spend on the postage when 8th set of letter is mailed.

Explanation

To solve the problem step by step, let's break it down clearly:

### Step 1: Understanding the Problem
A person writes a letter to four friends, and each of these friends then sends the letter to four more friends. This process continues, creating a chain of letters. We need to find out how many letters are sent by the time the 8th set of letters is mailed.

### Step 2: Determine the Pattern
1. The first person sends letters to 4 friends.
2. Each of those 4 friends sends letters to 4 more friends.
3. This creates a geometric progression (GP) where:
- The first term (number of letters sent by the first person) is \(4\).
- The second term (letters sent by the first set of friends) is \(4^2\).
- The third term is \(4^3\), and so on.

### Step 3: General Formula for the n-th Set of Letters
The number of letters sent in the n-th set can be expressed as:
\[ \text{Number of letters in n-th set} = 4^n \]
Thus, for the 8th set, we have:
\[ \text{Number of letters in 8th set} = 4^8 \]

### Step 4: Calculate \(4^8\)
Calculating \(4^8\):
\[ 4^8 = (2^2)^8 = 2^{16} = 65536 \]

### Step 5: Total Letters Sent Up to the 8th Set
To find the total number of letters sent up to the 8th set, we sum the letters from all sets:
\[ \text{Total letters} = 4^1 + 4^2 + 4^3 + ... + 4^8 \]
This is a geometric series where:
- First term \(a = 4\)
- Common ratio \(r = 4\)
- Number of terms \(n = 8\)

The sum of the first n terms of a geometric series is given by:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
Substituting the values:
\[ S_8 = 4 \frac{4^8 - 1}{4 - 1} = 4 \frac{65536 - 1}{3} = 4 \frac{65535}{3} \]

### Step 6: Calculate the Total Letters
Calculating:
\[ S_8 = 4 \cdot 21845 = 87380 \]

### Step 7: Calculate the Cost of Mailing Letters
The cost to mail one letter is 50 paise, which is equivalent to:
\[ \text{Cost per letter} = \frac{50}{100} = 0.5 \text{ rupees} \]
Thus, the total cost for mailing 87380 letters is:
\[ \text{Total cost} = 87380 \cdot 0.5 = 43690 \text{ rupees} \]

### Final Answer
The total amount spent on postage when the 8th set of letters is mailed is **43690 rupees**.

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