Key Concepts and Formulas

• Mean Deviation from the Mean (MD($\bar{x}$)): For a set of $n$ observations $x_1, x_2, \ldots, x_n$ with mean $\bar{x}$, the mean deviation from the mean is given by: $\text{MD}(\bar{x}) = \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}|$ This formula calculates the average of the absolute differences between each data point and the mean, providing a measure of data dispersion.
• Properties of an Arithmetic Progression (AP):
  • If a sequence $a, a+d, a+2d, \ldots, a+(n-1)d$ is an AP with $n$ terms, its mean ($\bar{x}$) can be calculated as the average of the first and last term: $\bar{x} = \frac{a_1 + a_n}{2}$.
  • The sum of the first $N$ natural numbers is $1+2+\ldots+N = \frac{N(N+1)}{2}$.

---

Step-by-Step Solution

1. Identify the Data Set and its Properties

• What we are doing: We are analyzing the given sequence of numbers to understand its structure and count the total number of terms.
• Why this step is important: Understanding the data structure (here, an Arithmetic Progression) allows us to use specific properties to simplify calculations for the mean and deviations.
• The given numbers are $1, 1+d, 1+2d, \ldots, 1+100d$.
  • This is an Arithmetic Progression (AP).
  • The first term is $a_1 = 1$.
  • The common difference is $d$.
  • The terms can be represented as $x_k = 1+kd$, where $k$ ranges from $0$ to $100$.
  • The total number of terms ($n$) is $100 - 0 + 1 = 101$.

2. Calculate the Mean ($\bar{x}$) of the Data Set

• What we are doing: Determining the central value of our data set, which is required for calculating deviations in the next step.
• Why this step is important: The Mean Deviation is defined with respect to the mean, so finding the mean is a prerequisite.
• For an AP, the mean is simply the average of the first and the last term:
  • First term ($a_1$) = $1$.
  • Last term ($a_n$) = $1+100d$. $\bar{x} = \frac{a_1 + a_n}{2} = \frac{1 + (1+100d)}{2} = \frac{2+100d}{2} = 1+50d$

3. Determine the Deviations from the Mean

• What we are doing: Calculating the difference between each data point and the mean, and then taking its absolute value.
• Why this step is important: The Mean Deviation formula specifically requires the sum of absolute deviations to ensure that positive and negative deviations do not cancel out.
• For each term $x_k = 1+kd$ (where $k=0, 1, \ldots, 100$), the deviation from the mean $\bar{x}=1+50d$ is: $x_k - \bar{x} = (1+kd) - (1+50d) = (k-50)d$
• The absolute deviation for each term is: $|x_k - \bar{x}| = |(k-50)d| = |k-50||d|$
• Observe the pattern of absolute deviations:
  • For $k=0$: $|0-50||d| = 50|d|$
  • For $k=1$: $|1-50||d| = 49|d|$
  • ...
  • For $k=49$: $|49-50||d| = |d|$
  • For $k=50$: $|50-50||d| = 0$ (This is the middle term, $x_{51}$, which equals the mean)
  • For $k=51$: $|51-50||d| = |d|$
  • ...
  • For $k=100$: $|100-50||d| = 50|d|$
• This shows a symmetric distribution of absolute deviations around the mean.

4. Sum the Absolute Deviations

• What we are doing: Summing up all the absolute deviations calculated in the previous step.
• Why this step is important: This sum forms the numerator of the Mean Deviation formula. Exploiting the symmetry of the AP significantly simplifies this summation.
• The sum of absolute deviations is $\sum_{k=0}^{100} |x_k - \bar{x}|$. Based on the pattern identified: $\sum_{k=0}^{100} |x_k - \bar{x}| = (50|d| + 49|d| + \ldots + |d|) + 0 + (|d| + \ldots + 49|d| + 50|d|)$ Due to symmetry, we can write this as: $\sum_{k=0}^{100} |x_k - \bar{x}| = 2 \times (|d| + 2|d| + \ldots + 50|d|)$
• Factor out $|d|$: $= 2|d| (1 + 2 + \ldots + 50)$
• Using the sum of the first $N$ natural numbers formula, $S_N = \frac{N(N+1)}{2}$, with $N=50$: $1 + 2 + \ldots + 50 = \frac{50(50+1)}{2} = \frac{50 \times 51}{2} = 25 \times 51 = 1275$
• Substitute this sum back: $\sum_{k=0}^{100} |x_k - \bar{x}| = 2|d| (1275) = 2550|d|$

5. Calculate the Mean Deviation from the Mean

• What we are doing: Applying the Mean Deviation formula using the total sum of absolute deviations and the number of terms.
• Why this step is important: This brings us closer to the final equation that will allow us to solve for $d$.
• Using the formula $\text{MD}(\bar{x}) = \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}|$: $\text{MD}(\bar{x}) = \frac{1}{101} (2550|d|) = \frac{2550}{101}|d|$

6. Equate to the Given Mean Deviation and Solve for $|d|$

• What we are doing: Using the problem's given Mean Deviation value to form an equation and solve for the unknown common difference $d$.
• Why this step is important: This is the final step in finding the value of $d$.
• We are given that $\text{MD}(\bar{x}) = 255$. $\frac{2550}{101}|d| = 255$
• To solve for $|d|$, multiply both sides by $\frac{101}{2550}$: $|d| = 255 \times \frac{101}{2550}$
• Notice that $2550 = 10 \times 255$. Simplify the expression: $|d| = \frac{255 \times 101}{10 \times 255} = \frac{101}{10}$ $|d| = 10.1$
• The question asks for "a value of d". Since $|d|=10.1$, $d$ can be $10.1$ or $-10.1$. Looking at the options, $10.1$ is provided.

---

Common Mistakes & Tips

• Forgetting Absolute Values: A crucial error is to omit the absolute value signs in the Mean Deviation formula. Without them, the sum of deviations from the mean will always be zero, leading to an incorrect result.
• Incorrect Term Count: Be careful when determining the number of terms ($n$) in an AP. If terms are indexed from $0$ to $K$ (like $1+0d, \ldots, 1+100d$), there are $K+1$ terms, not $K$.
• Leveraging AP Symmetry: For an Arithmetic Progression, the deviations from the mean exhibit symmetry. Recognizing and utilizing this symmetry (e.g., summing $1+2+\ldots+N$ and multiplying by 2) significantly simplifies the calculation of the sum of absolute deviations.
• Sum of Natural Numbers: Remember the formula for the sum of the first $N$ natural numbers, $\frac{N(N+1)}{2}$, as it frequently appears in such problems.

---

Summary

This problem required us to calculate the common difference of an Arithmetic Progression given its mean deviation from the mean. We began by identifying the properties of the AP, including the number of terms and its mean. The core of the solution involved carefully calculating the absolute deviation of each term from the mean, recognizing the inherent symmetry of these deviations. This symmetry allowed us to efficiently sum all absolute deviations. Finally, by equating the derived expression for the mean deviation to the given value, we successfully solved for $|d|$ and found a possible value for $d$.

The final answer is $\boxed{10.1}$, which corresponds to option (A).