Identify the data set

The numbers are:

Find the mean of the numbers

For an arithmetic progression, the mean is the middle term.

Since there are $101$ terms, the middle term is the $51$st term: $\bar{x}=1+50d$

Write the deviations from the mean

The terms are symmetric about the mean, so deviations are: $-50d,-49d,\dots,-d,0,d,\dots,49d,50d$

Hence the absolute deviations are: $50|d|,49|d|,\dots,|d|,0,|d|,\dots,49|d|,50|d|$

Compute the mean deviation from the mean

Mean deviation about the mean is: $\text{M.D.} = \frac{\text{sum of absolute deviations}}{101}$

Now, $\text{sum of absolute deviations} = 2|d|(1+2+\cdots+50)$

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

we get: $\text{sum of absolute deviations}=2|d|\cdot 1275=2550|d|$

Therefore, $\text{M.D.}=\frac{2550|d|}{101}$

Use the given condition

Given mean deviation is $255$: $\frac{2550|d|}{101}=255$

So, $2550|d| = 255\cdot 101$

$|d| = \frac{255\cdot 101}{2550} = \frac{101}{10}=10.1$

Thus, $d=\pm 10.1$

Match with the options

Among the given options, the listed value is: $\boxed{10.1}$

So the correct option is A.

Comparison with stored answer

Stored correct answer: A

Our derived answer: A

They agree.