The arithmetic mean of n numbers of a series is `bar(x)`. The sum of the first (n-1) numbers is k. The nth number is

Options

A. `n/2bar(x)-k`
B. `nbar(x)-k`
C. `1/nbar(x)-k`
D. none of these

Answer

To find the nth number in a series where the arithmetic mean of n numbers is denoted as \( \bar{x} \) and the sum of the first \( n-1 \) numbers is \( k \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Arithmetic Mean**:
The arithmetic mean \( \bar{x} \) of \( n \) numbers \( x_1, x_2, \ldots, x_n \) is given by the formula:
\[ \bar{x} = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} \]

2. **Expressing the Sum of n Numbers**:
We can express the sum of the \( n \) numbers as:
\[ x_1 + x_2 + \ldots + x_n = n \cdot \bar{x} \]

3. **Using the Given Information**:
We are given that the sum of the first \( n-1 \) numbers is \( k \). Thus, we can write:
\[ x_1 + x_2 + \ldots + x_{n-1} = k \]

4. **Finding the nth Number**:
The nth number, \( x_n \), can be found by rearranging the sum of all \( n \) numbers:
\[ x_n = (x_1 + x_2 + \ldots + x_n) - (x_1 + x_2 + \ldots + x_{n-1}) \]
Substituting the expressions we have:
\[ x_n = n \cdot \bar{x} - k \]

5. **Final Result**:
Therefore, the nth number is:
\[ x_n = n \bar{x} - k \]