The mean of 'n' observations is x. If the first observation is increased by 1. The second by 2, the third by 3 and so on, then the new mean is equal to :

Question

Accepted Answer

To determine the new mean when each observation is increased incrementally, we start by understanding the initial conditions and changes applied to the observations.

1. **Initial Mean Calculation:**
   - The mean of 'n' observations is given as $ \bar{x} $. This implies that the sum of all observations is $ n \times \bar{x} $.

2. **Changes to Observations:**
   - The first observation is increased by 1, the second by 2, the third by 3, and so on, up to the nth observation, which is increased by n.

3. **Total Increase in Sum:**
   - The total increase in the sum of observations due to these changes is the sum of the first n natural numbers: $ 1 + 2 + 3 + \ldots + n $.
   - The formula for the sum of the first n natural numbers is $ \frac{n(n+1)}{2} $.

4. **New Sum of Observations:**
   - The new sum of observations becomes $ n \times \bar{x} + \frac{n(n+1)}{2} $.

5. **New Mean Calculation:**
   - The new mean is the new sum of observations divided by the number of observations, n:
   $$
   \text{New Mean} = \frac{n \times \bar{x} + \frac{n(n+1)}{2}}{n}
   $$
   - Simplifying this expression:
   $$
   \text{New Mean} = \bar{x} + \frac{n(n+1)}{2n}
   $$
   $$
   \text{New Mean} = \bar{x} + \frac{n+1}{2}
   $$

6. **Conclusion:**
   - The new mean is $ \bar{x} + \frac{n+1}{2} $, which corresponds to Option 2.

**Explanation for Incorrect Options:**

- **Option 1: $ \bar{x} + (2n+1) $**
  - This option suggests a linear addition of $ 2n+1 $ to the mean, which does not account for the incremental increase pattern (1, 2, 3, ..., n) and is not derived from the sum of the first n natural numbers.

- **Option 3: $ \bar{x} + (n+1) $**
  - This option implies a constant addition of $ n+1 $ to the mean, which does not accurately reflect the arithmetic progression of the increases (1, 2, 3, ..., n).

- **Option 4: $ \bar{x} - \frac{(n-1)}{2} $**
  - This option incorrectly suggests a subtraction from the mean, which is not consistent with the fact that all observations are increased, leading to an increase in the mean.

Thus, Option 2 is the correct answer as it accurately represents the new mean after the specified changes to the observations.