The mean of the series a, a + d, a + 2d, …, a + 2 nd, is

Options

A. a + (n - 1) d
B. a + nd
C. a + (n + 1) d
D. none of these

Explanation

To find the mean of the series \( a, a + d, a + 2d, \ldots, a + 2nd \), we can follow these steps:

### Step-by-Step Solution

1. **Identify the series**: The series given is \( a, a + d, a + 2d, \ldots, a + 2nd \). This is an arithmetic progression (AP) where:
- First term \( (a) = a \)
- Common difference \( (d) = d \)
- Last term \( (l) = a + 2nd \)

2. **Determine the number of terms**: To find the number of terms in the series, we note that the last term can be expressed as:
\[ a + 2nd \]
The number of terms \( n \) can be calculated as follows:
- The first term is \( a \) (0th term).
- The last term is \( a + 2nd \), which corresponds to the \( 2n \)th term.
- Therefore, the total number of terms is \( 2n + 1 \).

3. **Calculate the sum of the series**: The sum \( S_n \) of the first \( n \) terms of an arithmetic progression can be calculated using the formula:
\[ S_n = \frac{n}{2} \times (a + l) \]
Here, substituting for \( n \) and \( l \):
\[ S = \frac{2n + 1}{2} \times \left( a + (a + 2nd) \right) \]
Simplifying this gives:
\[ S = \frac{2n + 1}{2} \times (2a + 2nd) = (2n + 1)(a + nd) \]

4. **Find the mean**: The mean \( M \) of the series is given by the total sum divided by the number of terms:
\[ M = \frac{S}{\text{Number of terms}} = \frac{(2n + 1)(a + nd)}{2n + 1} \]
The \( (2n + 1) \) cancels out:
\[ M = a + nd \]

### Final Answer
Thus, the mean of the series \( a, a + d, a + 2d, \ldots, a + 2nd \) is:
\[ \boxed{a + nd} \]