If the mean deviation of the numbers $$1, 1+d, \ldots, 1+100d$$ from their mean is 255, then a value of $$d$$ is:

The given sequence is an arithmetic progression with the first term $$a = 1$$ and common difference $$d$$. The terms are $$1, 1+d, 1+2d, \ldots, 1+100d$$. The number of terms is $$101$$ because it starts from $$k=0$$ (which gives $$1 + 0 \cdot d = 1$$) to $$k=100$$ (which gives $$1 + 100d$$).

To find the mean deviation, first compute the mean of the sequence. For an arithmetic progression, the mean is the average of the first and last terms. The first term is $$1$$ and the last term is $$1 + 100d$$, so:

Mean = $$\frac{1 + (1 + 100d)}{2} = \frac{2 + 100d}{2} = 1 + 50d$$.

The mean deviation (MD) is the average of the absolute deviations from the mean. The $$k$$-th term is $$x_k = 1 + kd$$, so the deviation from the mean is:

$$|x_k - \text{mean}| = |(1 + kd) - (1 + 50d)| = |kd - 50d| = |d| \cdot |k - 50|$$.

Since the options for $$d$$ are positive, assume $$d > 0$$, so $$|d| = d$$. Thus, $$|x_k - \text{mean}| = d \cdot |k - 50|$$.

The sum of absolute deviations is:

$$\sum_{k=0}^{100} |x_k - \text{mean}| = d \cdot \sum_{k=0}^{100} |k - 50|$$.

Compute the sum $$S = \sum_{k=0}^{100} |k - 50|$$. Split the sum into three parts: $$k$$ from 0 to 49, $$k = 50$$, and $$k$$ from 51 to 100.

• For $$k = 50$$, $$|k - 50| = 0$$.
• For $$k$$ from 0 to 49, $$|k - 50| = 50 - k$$.
• For $$k$$ from 51 to 100, $$|k - 50| = k - 50$$.

Thus, $$S = \sum_{k=0}^{49} (50 - k) + \sum_{k=51}^{100} (k - 50)$$.

Each sum has 50 terms. Compute the first sum:

$$\sum_{k=0}^{49} (50 - k) = (50 - 0) + (50 - 1) + \cdots + (50 - 49) = 50 + 49 + \cdots + 1$$.

This is the sum of the first 50 natural numbers: $$\frac{50 \times 51}{2} = 1275$$.

Similarly, the second sum:

$$\sum_{k=51}^{100} (k - 50) = (51 - 50) + (52 - 50) + \cdots + (100 - 50) = 1 + 2 + \cdots + 50 = \frac{50 \times 51}{2} = 1275$$.

Therefore, $$S = 1275 + 1275 = 2550$$.

The mean deviation is:

MD = $$\frac{1}{n} \times \text{sum of absolute deviations} = \frac{1}{101} \times d \cdot S = \frac{d \cdot 2550}{101}$$.

Given that MD = 255, set up the equation:

$$\frac{2550d}{101} = 255$$.

Solve for $$d$$:

Multiply both sides by 101: $$2550d = 255 \times 101$$.

Divide both sides by 255: $$d = \frac{255 \times 101}{2550}$$.

Simplify: $$d = \frac{101}{10} = 10.1$$.

Checking the options, 10.1 corresponds to Option A.

Hence, the correct answer is Option A.