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The arithmetic progression has first term 1 and common difference d. Its mean is given by [101 + d(1 + 2 + 3 +... + 100)]/101 = [1 + d × 100 × 101/(101 × 2)] = 1 + 50d. If the mean deviation about the mean is 255, then 1/101 [|1 − (1+50d)| + |(1+d) − (1+50d)| + |(1+2d) − (1+50d)| +... + |(1+100d) − (1+50d)|] = 255. Hence 2d[1 + 2 + 3 +... + 50] = 101 × 255, so 2d × 50 × 51/2 = 101 × 255, giving d = 101×255/(50×51) = 10.1.

1. (a)
2. (b)
3. (c)
4. (d)

Correct answer: (d)

Solution

The calculation shows that the common difference d is derived from the mean deviation formula, which simplifies to 2d multiplied by the sum of the first 50 integers equating to 101 times the mean deviation of 255. Solving this equation correctly leads to the value of d being 10.1, confirming option (d) as the correct answer.

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