Question

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

• A. 20.0
• B. 10.1
• C. 20.2
• D. 10.0

Hint:

Symmetric A.P.’s with odd terms use formula M.D. = \( \frac{2}{n}\sum_{k=1}^{(n-1)/2} k d \).

Answer 1

The Correct Option is D

Step 1: The sequence contains \(101\) terms in A.P. with first term \(a=1\) and last term \(l=1+100d\).
Step 2: Mean of an A.P.: \[ \bar a=\frac{a+l}{2}=\frac{2+100d}{2}=1+50d. \]
Step 3: Deviations from mean are symmetric about the central term. For an A.P. of odd number of equally spaced terms \(x_i=\bar a+(i-50)d\), mean deviation = average of \(|i-50|d\).
Step 4: \[ \text{M.D.}= \frac{2}{101}\sum_{k=1}^{50} k d =\frac{2}{101}\frac{50\cdot51}{2}d =\frac{2550}{101}d. \]
Step 5: Equate to 255: \[ \frac{2550}{101}d=255 \Rightarrow d=10. \] Hence → (D).