Fair-Asia is well known for its queer marketing practices. To lure its customers, out of the blue, it comes up with a very lucrative offer: In that offer it promises to sell and the air tickets, for a one-way trip to Bangkok that originates from Jakrata, in such a manner that the nth customer has to pay exactly Rs.n to buy an air ticket,but each prospective passenger has to bay a separate ticket for oneself. The airline claims that it had sold out all the seats at throm away prices in no time. However, at the last moment one of the customers had returned her ticket and got the full refund from the airline. Consequently, the average income of the airline from this particular flight is exactly rs 100 per passenger.
What's the loss incurred by the airline due to the refund of a ticket?

Options

A. rs199
B. rs100
C. rs 200
D. Cannot be determined

Answer

To solve the problem, we need to determine the loss incurred by the airline due to the refund of one ticket, given that the average income per passenger is Rs. 100.

### Step-by-Step Solution:

1. **Understanding the Average Income**:
The average income from the flight is given as Rs. 100 per passenger. This means that if there were 'n' passengers, the total income from all passengers would be \( 100n \).

**Hint**: Remember that the average is calculated by dividing the total income by the number of passengers.

2. **Calculating the Total Income**:
The nth customer pays Rs. n for the ticket. Therefore, the total income from 'n' passengers can be calculated using the formula for the sum of the first 'n' natural numbers:
\[ \text{Total Income} = 1 + 2 + 3 + ... + n = \frac{n(n + 1)}{2} \]

**Hint**: Use the formula for the sum of the first 'n' natural numbers to find the total income.

3. **Setting Up the Equation**:
Since we know the total income can also be expressed as \( 100n \), we can set up the equation:
\[ \frac{n(n + 1)}{2} = 100n \]

**Hint**: Equate the two expressions for total income to find 'n'.

4. **Solving for 'n'**:
Rearranging the equation gives:
\[ n(n + 1) = 200n \]
\[ n^2 + n - 200n = 0 \]
\[ n^2 - 199n = 0 \]
Factoring out 'n':
\[ n(n - 199) = 0 \]
Thus, \( n = 199 \) (since the number of passengers cannot be zero).

**Hint**: Factor the quadratic equation to find the possible values for 'n'.

5. **Calculating the Total Income with 199 Passengers**:
Now, substituting \( n = 199 \) back into the total income formula:
\[ \text{Total Income} = \frac{199(199 + 1)}{2} = \frac{199 \times 200}{2} = 19900 \]

**Hint**: Make sure to perform the multiplication and division carefully.

6. **Calculating the Refund Amount**:
If one passenger returns their ticket, the total number of passengers becomes \( 198 \). The average income remains Rs. 100, so the new total income must be:
\[ \text{New Total Income} = 100 \times 198 = 19800 \]

**Hint**: The average income helps to determine the new total income after one ticket is refunded.

7. **Finding the Loss**:
The loss incurred by the airline due to the refund of the ticket is the difference between the original total income and the new total income:
\[ \text{Loss} = \text{Original Total Income} - \text{New Total Income} = 19900 - 19800 = 100 \]

**Hint**: Subtract the new total income from the original total income to find the loss.

### Final Answer:
The loss incurred by the airline due to the refund of the ticket is Rs. 100.