Mean of n observations is `barx.` If each of these observations is increased by 1, 2, 3, 4...n, respectively, then what will be their mean?

Answer

To solve the problem step by step, we need to find the new mean after increasing each of the n observations by a sequence of numbers (1, 2, 3, ..., n).

### Step-by-Step Solution:

1. **Understand the Mean**:
The mean of n observations is given as \( \bar{x} \). This means:
\[ \bar{x} = \frac{\Sigma x}{n} \]
where \( \Sigma x \) is the sum of all observations.

2. **Express the Sum of Observations**:
From the mean, we can express the sum of the observations as:
\[ \Sigma x = n \cdot \bar{x} \]

3. **Increase Each Observation**:
Each observation is increased by 1, 2, 3, ..., n respectively. So, the new observations become:
\[ x_1 + 1, \quad x_2 + 2, \quad x_3 + 3, \quad \ldots, \quad x_n + n \]

4. **Calculate the New Sum**:
The new sum of the observations can be calculated as:
\[ \Sigma x' = (x_1 + 1) + (x_2 + 2) + (x_3 + 3) + \ldots + (x_n + n) \]
This can be rewritten as:
\[ \Sigma x' = \Sigma x + (1 + 2 + 3 + \ldots + n) \]
The sum of the first n natural numbers is given by the formula:
\[ 1 + 2 + 3 + \ldots + n = \frac{n(n + 1)}{2} \]
Therefore, we can write:
\[ \Sigma x' = n \cdot \bar{x} + \frac{n(n + 1)}{2} \]

5. **Calculate the New Mean**:
The new mean \( \bar{x}' \) can be calculated as:
\[ \bar{x}' = \frac{\Sigma x'}{n} \]
Substituting the expression for \( \Sigma x' \):
\[ \bar{x}' = \frac{n \cdot \bar{x} + \frac{n(n + 1)}{2}}{n} \]
This simplifies to:
\[ \bar{x}' = \bar{x} + \frac{n + 1}{2} \]

6. **Final Result**:
Thus, the new mean after increasing each observation by 1, 2, 3, ..., n is:
\[ \bar{x}' = \bar{x} + \frac{n + 1}{2} \]