If the mean deviation of the number 1,1+d,1+2d ,.., 1 + 100 d from their mean is 255 then d is equal to

Options

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

Answer

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 255. Let's break this down step by step.

### Step 1: Identify the terms and the number of terms
We have the sequence:
\[ 1, 1+d, 1+2d, \ldots, 1+100d \]
This is an arithmetic progression (AP) with:
- First term \( a = 1 \)
- Common difference \( d \)
- Number of terms \( n = 101 \) (from \( 1 \) to \( 100d \))

### Step 2: Calculate the mean of the sequence
The mean \( \bar{x} \) of an AP can be calculated using the formula:
\[ \bar{x} = \frac{\text{Sum of terms}}{n} \]
The sum of the first \( n \) terms of an AP is given by:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
Substituting the values:
\[ S_{101} = \frac{101}{2} \times \left(2 \cdot 1 + (101 - 1)d\right) = \frac{101}{2} \times (2 + 100d) \]
Thus, the mean is:
\[ \bar{x} = \frac{S_{101}}{101} = \frac{101}{101} \times \frac{(2 + 100d)}{2} = 1 + 50d \]

### Step 3: Calculate the mean deviation
The mean deviation \( MD \) is given by:
\[ MD = \frac{\sum |x_i - \bar{x}|}{n} \]
Where \( x_i \) are the terms of the sequence. We need to calculate \( |x_i - \bar{x}| \):
- For \( i = 0 \): \( |1 - (1 + 50d)| = | - 50d | = 50d \)
- For \( i = 1 \): \( |(1 + d) - (1 + 50d)| = | - 49d | = 49d \)
- For \( i = 2 \): \( |(1 + 2d) - (1 + 50d)| = | - 48d | = 48d \)
- Continuing this way, we find that:
\[ |x_i - \bar{x}| = |(1 + id) - (1 + 50d)| = |(i - 50)d| \]
Thus, we have:
\[ |x_0 - \bar{x}| + |x_1 - \bar{x}| + \ldots + |x_{100} - \bar{x}| = 50d + 49d + 48d + \ldots + 0 + \ldots + 49d + 50d \]
This is the sum of the first 50 natural numbers multiplied by \( d \):
\[ = 2(50 + 49 + \ldots + 1)d = 2 \cdot \frac{50 \cdot 51}{2}d = 2550d \]

### Step 4: Set up the equation for mean deviation
Now we can set up the equation for the mean deviation:
\[ MD = \frac{2550d}{101} = 255 \]
Multiplying both sides by 101:
\[ 2550d = 255 \times 101 \]
Calculating the right side:
\[ 255 \times 101 = 25755 \]
Thus:
\[ 2550d = 25755 \]
Dividing both sides by 2550:
\[ d = \frac{25755}{2550} = 10.1 \]

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