Question

If the mean deviation of the data: $$ 1, 1 + d, 1 + 2d, \ldots, 1 + 100d $$ from their mean is 255, then find $ d $.

• A. 10.1
• B. 10.2
• C. 10.3
• D. 10.4

Hint:

Mean deviation of an AP with odd terms about the mean is: \( \frac{d}{n} \sum |x_i - \text{mean}| \), which simplifies using symmetry.

Answer 1

The Correct Option is A

The sequence is an arithmetic progression (AP) with: - First term: \( a = 1 \) - Common difference: \( d \) - Number of terms: 101 (from \( a \) to \( a + 100d \)) Mean of the sequence: \[ \text{Mean} = \frac{1 + (1 + 100d)}{2} = \frac{2 + 100d}{2} = 1 + 50d \] Mean deviation of an AP with odd number of terms = \[ \frac{d}{n} \cdot \sum_{k=1}^{n} |k - \frac{n+1}{2}| \quad \text{(deviation from mean)} \] But since AP is symmetric, mean deviation from the mean for \( n = 101 \) terms: \[ \text{Mean Deviation} = \frac{2d}{101} \cdot \sum_{k=1}^{50} k = \frac{2d}{101} \cdot \frac{50 \cdot 51}{2} = \frac{d}{101} \cdot 2550 = \frac{2550d}{101} \] Set equal to 255: \[ \frac{2550d}{101} = 255 \Rightarrow d = \frac{255 \cdot 101}{2550} = \frac{101}{10} = \boxed{10.1} \]