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

Q: Wait, why is the number of terms $n=101$ and not $100$? The last term is $1+100d$.

This is a classic trap! Think of the sequence as $1+0d, 1+1d, 1+2d, \dots, 1+100d$. The coefficient of $d$ ranges from $0$ to $100$. If you count the integers from $0$ to $100$ inclusive, you get $100 - 0 + 1 = 101$ terms. Always check the starting index!

Q: Why can't we just use the standard formula for Mean Deviation? It seems like a lot of work.

You absolutely can, but the standard formula $\frac{1}{n} \sum |x_i - \bar{x}|$ is a summation. In an AP, the symmetry allows us to pair terms equidistant from the mean, which turns a complex summation into a simple arithmetic series. It is not just faster; it is a deeper way to understand the structure of the data.

Q: Why does the absolute value disappear in the summation step?

The absolute value doesn't disappear; it is handled by the symmetry. Since the mean is the middle term ($1+50d$), the terms before the mean are $1+kd$ where $k 50$. The distance $|k-50|d$ is the same for $k=50-m$ and $k=50+m$. By grouping these symmetric pairs, we effectively remove the absolute value sign by multiplying the sum of the first $50$ integers by $2$.

Question

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

Options:

(A) 20.0

(B) 10.1

(C) 20.2

(D) 10.0

Sigma Percentile · Accepted Answer

### Analyzing the Sequence Anatomy

First, let us count our markers. It is a common pitfall to look at the last term, $1+100d$, and assume there are $100$ terms.

But look closer at the coefficients of $d$: they start at $0$ (for the first term $1$) and end at $100$. Counting from $0$ to $100$ gives us exactly $n=101$ terms. This is our first victory.

### Finding the Center

In any symmetric sequence like an Arithmetic Progression (AP), the mean $\bar{x}$ is the exact middle. The formula for the mean is:

$$\bar{x} = \frac{	ext{First term} + 	ext{Last term}}{2}$$

Substituting our values, we get:

$$\bar{x} = \frac{1 + (1 + 100d)}{2} = \frac{2 + 100d}{2} = 1 + 50d$$

This value, $1 + 50d$, represents the heart of our sequence and the balance point.

### The Geometry of Deviation

The Mean Deviation ($MD$) is defined as:

$$MD = \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}|$$

For any term $x_k = 1 + kd$, the deviation from the mean is:

$$|(1+kd) - (1+50d)| = |k-50|d$$

This tells us that the distance of each point from the mean is simply the distance of its index $k$ from $50$, scaled by $d$.

### The Symmetry Trick

Now, let us sum these deviations. We need to calculate:

$$d 	imes \sum_{k=0}^{100} |k-50| = d 	imes (|0-50| + |1-50| + \dots + |49-50| + |50-50| + |51-50| + \dots + |100-50|)$$

Notice the beautiful symmetry! The terms $|0-50|$ and $|100-50|$ are both $50$. The terms $|1-50|$ and $|99-50|$ are both $49$. This continues until we reach the middle term, which is $0$.

We can rewrite this sum as:

$$2d 	imes (50 + 49 + \dots + 1)$$

Using the sum of the first $n$ natural numbers formula, $\frac{n(n+1)}{2}$, where $n=50$, we get:

$$2d 	imes \frac{50 	imes 51}{2} = 2550d$$

### Final Calculation

We are given that the Mean Deviation is $255$. So, we set up our final equation:

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

Dividing both sides by $255$, we get:

$$\frac{10d}{101} = 1$$

This leads us directly to $10d = 101$, which yields the final result:

**$d = 10.1$**

Animated Solution for Mathematics - Statistics: If the mean deviation of the numbers 1, 1 + d, 1 + 2d, .... 1 + 100d from their mean is 255, then d is equal to:

Select Answer:

Visualized Solution

Visualizing the Sequence

Finding the Total Number of Terms $n$

Calculating the Mean $\overset{x}{ˉ}$

Simplifying the Mean Expression

Defining Mean Deviation

Calculating Individual Deviations

Grouping the Symmetric Deviations

Evaluating the Sum

Substituting into the Formula

Solving for $d$

Final Conclusion

The Sigma Insight: Mean Deviation

Analyzing the Sequence Anatomy

Finding the Center

The Geometry of Deviation

The Symmetry Trick

Final Calculation

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