If AM of numbers `x_(1),x_(2)…..x_(n)` is `mu`, then what is tha AM of the numbers which are increased by 1, 2, 3, …..n respectively?

Options

A. `mu+((n+1)/(2))`
B. `mu`
C. `mu+(n(n+1))/(2)`
D. `mu-((n+1)/(2))`

Explanation

To solve the problem, we need to find the Arithmetic Mean (AM) of the new set of numbers formed by increasing each of the original numbers \( x_1, x_2, \ldots, x_n \) by \( 1, 2, 3, \ldots, n \) respectively.

### Step-by-Step Solution:

1. **Understand the Given Information**:
We are given that the Arithmetic Mean (AM) of the numbers \( x_1, x_2, \ldots, x_n \) is \( \mu \). This can be expressed mathematically as:
\[ \mu = \frac{x_1 + x_2 + \ldots + x_n}{n} \]
From this, we can derive the sum of the original numbers:
\[ x_1 + x_2 + \ldots + x_n = n \mu \]

2. **Form the New Set of Numbers**:
The new numbers are formed by adding \( 1, 2, 3, \ldots, n \) to \( x_1, x_2, \ldots, x_n \) respectively. Thus, the new numbers can be expressed as:
\[ (x_1 + 1), (x_2 + 2), (x_3 + 3), \ldots, (x_n + n) \]

3. **Calculate the Sum of the New Numbers**:
The sum of the new numbers can be calculated as follows:
\[ \text{Sum of new numbers} = (x_1 + 1) + (x_2 + 2) + (x_3 + 3) + \ldots + (x_n + n) \]
This can be rewritten as:
\[ = (x_1 + x_2 + \ldots + x_n) + (1 + 2 + 3 + \ldots + n) \]
We already know that \( x_1 + x_2 + \ldots + x_n = n \mu \). The sum of the first \( n \) natural numbers is given by the formula:
\[ 1 + 2 + 3 + \ldots + n = \frac{n(n + 1)}{2} \]
Therefore, the total sum of the new numbers becomes:
\[ n \mu + \frac{n(n + 1)}{2} \]

4. **Calculate the AM of the New Numbers**:
The AM of the new numbers is given by the total sum divided by the number of observations, which is \( n \):
\[ \text{AM of new numbers} = \frac{n \mu + \frac{n(n + 1)}{2}}{n} \]
Simplifying this expression:
\[ = \mu + \frac{n + 1}{2} \]

### Final Result:
Thus, the Arithmetic Mean of the new numbers is:
\[ \text{AM} = \mu + \frac{n + 1}{2} \]