If the mean deviation of numbers 1, 1+d, 1+2d, , 1+100d from their mean is 255 , then d is equal to: - Mathematics MCQ AIEEE 2009 2026 | ExamTest.live

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

Solution & Step-by-step Explanation

The numbers are in AP: a, a + d, \dots, a + n d where n = 100 .Total terms = 101 .Mean \overset{x}{ˉ} = \frac{a + ( a + n d )}{2} = \frac{1 + 1 + 100 d}{2} = 1 + 50 d .Mean Deviation (M.D.) = \frac{1}{N} \sum ∣ x_{i} - \overset{x}{ˉ} ∣ . M.D. = \frac{1}{101} [∣1 - (1 + 50 d) ∣ + ∣ (1 + d) - (1 + 50 d) ∣ + \dots + ∣ (1 + 100 d) - (1 + 50 d) ∣] M.D. = \frac{d}{101} [50 + 49 + \dots + 1 + 0 + 1 + \dots + 50] = \frac{d}{101} \cdot 2 \cdot \frac{50 \times 51}{2} M.D. = \frac{50 \times 51 \times d}{101} = 255 . 2550 d = 255 \times 101 ⟹ 10 d = 101 ⟹ d = 10.1 .

Try it yourself before checking the explanation above.

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

10.0

20.0

10.1

20.2

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