The AM of n numbers of a series is `bar(X). ` If the sum of first (n-1) terms is k, then the `n^(th)` number is

Options

A. `bar(X)-k`
B. `n bar(X)-k`
C. `bar(X)-n k`
D. `n bar(X)-n k`

Answer

To find the nth number in a series given the arithmetic mean (AM) and the sum of the first (n-1) terms, we can follow these steps:

### Step-by-Step Solution:

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

2. **Express the Sum of the Series**:
Let the sum of the first (n-1) terms be \( k \). Therefore, we can express the sum of all n terms as:
\[ \text{Sum of n terms} = k + x_n \]
where \( x_n \) is the nth number we want to find.

3. **Set Up the Equation Using the AM**:
From the definition of AM, we have:
\[ \bar{X} = \frac{k + x_n}{n} \]

4. **Cross-Multiply to Eliminate the Denominator**:
By cross-multiplying, we get:
\[ n \bar{X} = k + x_n \]

5. **Rearrange to Solve for \( x_n \)**:
Now, isolate \( x_n \):
\[ x_n = n \bar{X} - k \]

### Final Answer:
Thus, the nth number \( x_n \) is given by:
\[ x_n = n \bar{X} - k \]