If the mean deviation of the numbers `1,""1""+""d ,""1""+""2d ,""..."",""1""+""100 d` from their mean is 25, then the d is equal to
(1) 10.0
(2) 20.0
(3) 10.1
(4) 20.2

Options

A. `10.0`
B. `20.0`
C. 10.1
D. 20.2

Explanation

To solve the problem, we need to find the value of \( d \) given that the mean deviation of the numbers \( 1, 1 + d, 1 + 2d, \ldots, 1 + 100d \) from their mean is 25.

### Step 1: Calculate the Mean
The numbers can be represented as:
- First term: \( 1 \)
- Last term: \( 1 + 100d \)
- Total number of terms, \( n = 101 \)

The mean \( \bar{x} \) is calculated as:
\[ \bar{x} = \frac{\text{Sum of all numbers}}{n} \]
The sum of an arithmetic series can be calculated as:
\[ \text{Sum} = \frac{n}{2} \times (\text{First term} + \text{Last term}) = \frac{101}{2} \times (1 + (1 + 100d)) = \frac{101}{2} \times (2 + 100d) = 101 + 5050d \]
Thus, the mean is:
\[ \bar{x} = \frac{101 + 5050d}{101} = 1 + 50d \]

### Step 2: Set Up the Mean Deviation Equation
The mean deviation about the mean is given as 25. The formula for mean deviation \( MD \) is:
\[ MD = \frac{1}{n} \sum |x_i - \bar{x}| \]
Substituting the known values:
\[ 25 = \frac{1}{101} \sum |(1 + kd) - (1 + 50d)| \quad \text{for } k = 0, 1, 2, \ldots, 100 \]
This simplifies to:
\[ 25 = \frac{1}{101} \sum |kd - 50d| = \frac{1}{101} \sum |d(k - 50)| \]
This can be rewritten as:
\[ 25 = \frac{d}{101} \sum |k - 50| \]

### Step 3: Calculate the Summation
The values of \( k \) range from 0 to 100. The absolute deviation from 50 can be calculated as:
- For \( k = 0 \) to \( 50 \): \( |k - 50| = 50 - k \)
- For \( k = 51 \) to \( 100 \): \( |k - 50| = k - 50 \)

Calculating the sum:
\[ \sum_{k=0}^{50} (50 - k) + \sum_{k=51}^{100} (k - 50) \]
The first part:
\[ \sum_{k=0}^{50} (50 - k) = 50 + 49 + 48 + \ldots + 0 = \frac{50 \times 51}{2} = 1275 \]
The second part:
\[ \sum_{k=51}^{100} (k - 50) = 1 + 2 + \ldots + 50 = \frac{50 \times 51}{2} = 1275 \]
Thus, the total sum is:
\[ \sum |k - 50| = 1275 + 1275 = 2550 \]

### Step 4: Substitute Back into the Mean Deviation Equation
Now substituting back:
\[ 25 = \frac{d}{101} \times 2550 \]
Solving for \( d \):
\[ 25 \times 101 = 2550d \implies 2525 = 2550d \implies d = \frac{2525}{2550} = 10.1 \]

### Conclusion
Thus, the value of \( d \) is:
\[ \boxed{10.1} \]