Problem 52 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If the mean deviation of the numbers \(1,1+\mathrm{d}, \ldots, 1+100 \mathrm{~d}\) from their mean is 255 , then a value of \(\mathrm{d}\) is : [Online April 9, 2016] (a) \(10.1\) (b) \(5.05\) (c) \(20.2\) (d) 10

Step-by-Step Solution

Answer

The value of \(d\) is 10.1.

Step 1: Understand the Problem

We need to find the value of \(d\) for the sequence \(1, 1+d, 1+2d, \ldots, 1+100d\) given that the mean deviation of these numbers from their mean is 255.

Step 2: Calculate the Mean of the Sequence

The sequence contains 101 terms. The last term is \(1+100d\). The mean \(M\) of the sequence is given by:\[ M = \frac{1 + (1 + 100d)}{2} = \frac{2 + 100d}{2} = 1 + 50d.\]

Step 3: Find the Mean Deviation Formula

The mean deviation (MD) from the mean for the sequence is given as sum of absolute deviations divided by number of terms:\[ MD = \frac{1}{101} \sum_{i=0}^{100} |(1 + id) - (1 + 50d)|.\] This simplifies to:\[ MD = \frac{1}{101} \sum_{i=0}^{100} |id - 50d| = \frac{1}{101} \sum_{i=0}^{100} d|i - 50|.\]

Step 4: Simplification of Mean Deviation

Simplifying the absolute deviation sum for two cases - when \(i \leq 50\) and when \(i > 50\), the sums become symmetric. Both parts will have an equal sum, so it's sufficient to calculate from \(0\) to \(50\) and then multiply by 2. Thus: \[\sum_{i=0}^{100}|i - 50| = 2 \times \sum_{i=0}^{50} (50-i).\]

Step 5: Calculate Sum of Differences

The sum \(\sum_{i=0}^{50} (50-i)\) is a simple arithmetic series, calculated as follows:\[\sum_{i=0}^{50} (50-i) = \sum_{i=0}^{50} i = \frac{50 \times 51}{2} = 1275.\] Therefore, the total sum for all differences is \(2 \times 1275 = 2550.\)

Step 6: Find d Using Mean Deviation

Given the MD formula from Step 3, \[\text{MD} = \frac{d \times 2550}{101} = 255.\] Solving for \(d\), \[d = \frac{255 \times 101}{2550} = 10.1.\]

Step 7: Value Selection from Given Options

Among the options, \(d = 10.1\) corresponds to option (a).

Key Concepts

Arithmetic Sequences

Arithmetic sequences are a type of mathematical sequence where the difference between consecutive terms is constant. This difference is called the "common difference" and is typically represented by the letter \(d\). For example, in the sequence given in the exercise, which is \(1, 1+d, 1+2d, \ldots, 1+100d\), the common difference \(d\) is the same throughout.

• The first term of an arithmetic sequence is usually denoted as \(a\).
• The \(n\)-th term \(a_n\) can be expressed as \(a + (n-1)d\).

In our example, the first term \(a\) is 1 and there are 101 terms in total with the last term being \(1 + 100d\). Understanding this concept allows us to find any term in the sequence using the formula for the \(n\)-th term.

Mean of a Sequence

The mean of a sequence, also known as the average, is simply the sum of all terms divided by the number of terms. For our arithmetic sequence \(1, 1+d, 1+2d, \ldots, 1+100d\), finding the mean can be straightforward if you know the general formula:

• First, find the mean formula: \( M = \frac{1 + (1 + 100d)}{2}\)

This gives us the mean \(M = 1 + 50d\), which is derived from the properties of arithmetic sequences, where the mean is equivalent to the average of the first and last terms.
Calculating the mean is crucial because it becomes a reference point for measuring how "spread out" the sequence's terms are from this central value.

Absolute Deviation

Absolute deviation refers to the absolute differences between each term in a data set and a central value, typically the mean. The mean deviation, in particular, is the average of these absolute deviations and provides a measure of dispersion within the sequence.

• For instance, in our sequence \(1, 1+d, 1+2d, \ldots, 1+100d\), the mean is \(1 + 50d\).
• The deviations are calculated as \( |x_i - (1 + 50d)| \) where \(x_i\) are the sequence terms.

We sum these deviations and divide by the total number of terms. For our sequence, the mean deviation (MD) is given as 255, which translates mathematically to:
\[ \text{MD} = \frac{1}{101} \sum_{i=0}^{100} |i d - 50d| \] Understanding absolute deviation helps in grasping how widely the terms are distributed around the mean, indicating the variability within the sequence.