1. Identify the data set

   The numbers are in arithmetic progression: $1,\, 1+d,\, 1+2d,\, \dots,\, 1+100d$

   So there are $101$ terms.

2. Find the mean

   For an arithmetic progression, the mean is the average of the first and last terms: $\bar{x} = \frac{1+(1+100d)}{2} = 1+50d$

3. Write deviations from the mean

   The general term is $x_k = 1+kd \quad (k=0,1,2,\dots,100)$

   Its deviation from the mean is $x_k-\bar{x} = (1+kd)-(1+50d) = (k-50)d$

   Hence the absolute deviation is $|x_k-\bar{x}| = |k-50|\,|d|$

4. Compute the mean deviation from the mean

   By definition, $\text{M.D.} = \frac{1}{101}\sum_{k=0}^{100}|x_k-\bar{x}|$

   So, $255 = \frac{|d|}{101}\sum_{k=0}^{100}|k-50|$

5. Evaluate the sum

   The values of $|k-50|$ for $k=0$ to $100$ are: $50,49,48,\dots,1,0,1,\dots,48,49,50$

   Therefore, $\sum_{k=0}^{100}|k-50| = 2(1+2+\cdots+50)$

   Now, $1+2+\cdots+50 = \frac{50\cdot 51}{2} = 1275$

   Hence, $\sum_{k=0}^{100}|k-50| = 2\times 1275 = 2550$

6. Solve for $d$

   Substitute into the mean deviation formula: $255 = \frac{|d|}{101}\times 2550$

   $|d| = \frac{255\cdot 101}{2550}$

   Since $2550=255\times 10$, $|d| = \frac{101}{10} = 10.1$

   Thus, $d=10.1$ (taking the positive value from the options).

7. Check options

   • A: $20.0$ ✗
   • B: $10.1$ ✓
   • C: $20.2$ ✗
   • D: $10.0$ ✗

Final Answer: Option B.