Problem 24 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the mean of a set of observations \(x_{1}, x_{2}, x_{3}, \ldots, x_{n}\) is \(\bar{x}\), then mean of observations \(x_{i}+3 i \forall i=1,2,3, \ldots n\) equals (A) \(\bar{x}+3(n+1)\) (B) \(\bar{x}+\frac{3(n+1)}{2}\) (C) \(\bar{x}+\frac{n+1}{2 n}\) (D) None of these

Step-by-Step Solution

Answer

(B) \(\bar{x} + \frac{3(n+1)}{2}\)

Step 1: Understanding the Problem

We start with a set of observations \(x_1, x_2, x_3, \ldots, x_n\) with a given mean \(\bar{x}\). We need to find the mean of a new set of observations formed by adding \(3i\) to each \(x_i\) (i.e., \(x_i + 3i\) for each observation).

Step 2: Recall the Formula for Mean

The mean of a set of observations \(x_1, x_2, x_3, \ldots, x_n\) is given by \(\bar{x} = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n}\). This states that the mean is the total sum of the observations divided by the number of observations \(n\).

Step 3: Form the Sum of New Observations

For the new observations \(x_1 + 3 \cdot 1, x_2 + 3 \cdot 2, x_3 + 3 \cdot 3, \ldots, x_n + 3n\), their sum is: \((x_1 + x_2 + x_3 + \ldots + x_n) + (3 \cdot 1 + 3 \cdot 2 + 3 \cdot 3 + \ldots + 3n)\).

Step 4: Calculate the Additional Sum

The additional sum \(3 \cdot 1 + 3 \cdot 2 + 3 \cdot 3 + \ldots + 3n\) can be factored as \(3(1 + 2 + 3 + \ldots + n)\). The sum \(1 + 2 + 3 + \ldots + n\) is a known arithmetic series given by \(\frac{n(n+1)}{2}\).

Step 5: Calculate the New Mean

Substituting for the sum, the new total sum is \((x_1 + x_2 + x_3 + \ldots + x_n + 3 \cdot \frac{n(n+1)}{2})\). The mean of the new observations then is \(\frac{\text{Sum of New Observations}}{n} = \frac{(x_1 + x_2 + x_3 + \ldots + x_n) + \frac{3n(n+1)}{2}}{n}\).

Step 6: Simplify the Expression

Simplifying the expression, the new mean becomes \(\frac{x_1 + x_2 + \ldots + x_n}{n} + \frac{3n(n+1)}{2n}\). Recognizing that the first term is \(\bar{x}\), the new mean becomes \(\bar{x} + \frac{3(n+1)}{2}\).

Step 7: Choose the Correct Option

Comparing our result \(\bar{x} + \frac{3(n+1)}{2}\) with the given options, the correct answer is option (B).

Key Concepts

Arithmetic Series

An arithmetic series is a sequence of numbers in which each term after the first is obtained by adding a constant difference to the previous term. This constant difference is known as the 'common difference.' A series can be represented as:

• The first term: \(a\)
• The common difference: \(d\)

The general form of an arithmetic series can be written as \(a, a+d, a+2d, \, \ldots\), and so on. For example, if you have the numbers 1, 4, 7, 10, ..., the common difference here is 3. The sum of the first \(n\) terms of an arithmetic series can be calculated using the formula:\[S_n = \frac{n}{2} \times (2a + (n-1)d)\] In this exercise, adding \(3i\) to each observation creates a series of numbers that form an arithmetic sequence. The sum of these differences is used to adjust the overall mean of the observations.

Arithmetic Mean

The arithmetic mean is one of the most common ways to represent a set of observations through a single value that summarizes the entire dataset. Technically, it is the sum of the values in a dataset divided by the number of values: \[\bar{x} = \frac{x_1 + x_2 + \ldots + x_n}{n}\] This formula calculates what is often referred to as the average. It provides a balance point for the data, offering a simple snapshot of the overall dataset.
To understand this in practice, consider a set of values like 2, 4, 6. The arithmetic mean would be \[\frac{2+4+6}{3} = 4\] In problems like the one in our exercise, the aim is to analyze how adding consistent changes, like \(3i\) to each observation, affects this mean.

Mathematical Problem Solving

Mathematical problem-solving is a fundamental skill that helps in understanding and interacting with complex problems systematically. Here is a typical approach to solving mathematical problems:

• Understand the Problem: Identify what's given and what's needed. In the exercise, we start with a set of observations and aim to find a modified mean.
• Formulate a Plan: Recall relevant formulas and principles—including the concepts of arithmetic series and mean.
• Carry Out the Plan: Execute your strategy—calculate the additional sum with added modifications.
• Review and Check: Ensure your solution is logical and matches expectations—in this case, selecting the correct mean transformation option.

These steps can turn a seemingly complex problem into an easier and more manageable process. Just like in our exercise, understanding how changes in data affect the arithmetic mean is a cycle of recognizing patterns, simplifying complex elements, and validating the results.