If the mean deviation of the number \(1,1+\mathrm{d}, 1+2 \mathrm{~d}, \ldots \ldots 1+50 \mathrm{~d}\). From their mean is 260 then d is (a) \(20.5\) (b) \(20.3\) (c) \(20.4\) (d) \(10.4\)

The list of numbers starts with 1 and ends with \(1+50d\). There are 51 terms in this list. To compute the mean, which we will represent as \(\mu\), we need to add all the terms and divide by 51: \[\mu = \frac{1 + (1+d) + (1+2d) + \ldots + (1+50d)}{51}\] Notice that each term in the sum contains 1, so we can rewrite this as: \[\mu = \frac{51+ (d + 2d + \ldots + 50d)}{51}\] To find the sum of the terms \(d + 2d + \ldots + 50d\), we factor out \(d\): \(d (1 + 2 + \ldots + 50)\) Now find the sum of the numbers from 1 to 50 and substitute for the sum in the expression: \(d \cdot \frac{50 \cdot 51}{2}\) Now substitute back in to the expression for \(\mu\): \[\mu = \frac{51+ (d \cdot \frac{50 \cdot 51}{2})}{51}\] Simplify the expression to get the mean in terms of \(d\): \[\mu = 1 + \frac{25d}{1}\]

To calculate the mean deviation of the numbers from their mean, we take the absolute value of the difference between each number and the mean, and sum these up. Then we divide by the number of terms, which is 51. Representing the mean deviation as \(MD\), we have: \[MD = \frac{|(1 - \mu)| + |(1+d - \mu)| + |(1+2d - \mu)| + \ldots + |(1+50d - \mu)|}{51}\]

Since we are given the mean deviation as 260, we can set the equation: \[260 = \frac{|(1 - \mu)| + |(1+d - \mu)| + |(1+2d - \mu)| + \ldots + |(1+50d - \mu)|}{51}\] Now, substitute the value of \(\mu\) that we found in step 1: \[260 = \frac{|(1 - (1 + 25d))| + |(1+d - (1 + 25d))| +|(1+2d-(1 + 25d))| + \ldots + |(1+50d - (1 + 25d))|}{51}\] Simplify the equation: \[260 = \frac{|(-25d)| + |(-24d)| + |(-23d)| + \ldots + |(25d)|}{51}\] We know that the sum of the absolute values will be positive, so we can remove the absolute value signs and multiply the equation by 51 to isolate our sum: \[(260)(51) = 2(1 + 2 + 3 + \ldots + 25) \cdot d\] Now, we know the sum of numbers from 1 to 25, which we can substitute in the above equation: \[(260)(51) = 2 \cdot \frac{25 \cdot 26}{2}d\] Simplify and solve for \(d\): \[260 \cdot 51 = 25 \cdot 26 \cdot d\] \[d = \frac{260 \cdot 51}{25 \cdot 26}\] After evaluating the formula, we get \(d \approx 20.4\). So, the correct answer is: (c) \(20.4\)