Problem 1 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The average of \(n\) numbers \(x_{1}, x_{2}, x_{3}, \ldots, x_{n}\) is \(M .\) If \(x_{n}\) is replaced by \(x^{\prime}\), then new average is (A) \(M-x_{n}+x^{\prime}\) (B) \(\frac{n M-x_{n}+x^{\prime}}{n}\) (C) \(\frac{(n-1) M+x^{\prime}}{m}\) (D) \(\frac{M-x_{n}+x^{\prime}}{n}\)

Step-by-Step Solution

Answer

The correct answer is (B) \(\frac{nM-x_{n}+x^{\prime}}{n}\).

Step 1: Understand the Problem

We want to find the new average when the last number in a sequence is replaced with a different number. We are comparing different mathematical expressions to find the correct new average formula.

Step 2: Express the Initial Average

The average of the initial set of numbers is given by the formula \(M = \frac{x_1 + x_2 + x_3 + \cdots + x_n}{n}\).

Step 3: Calculate the Total Sum Using the Average

Rearrange the formula for the average to find the total sum: \(nM = x_1 + x_2 + x_3 + \cdots + x_n\).

Step 4: Substitute the Number

To find the new sum when \(x_n\) is replaced by \(x'\), use \(nM - x_n + x'\).

Step 5: Calculate the New Average

Divide the new total sum by \(n\) to find the new average: \( \frac{nM - x_n + x'}{n} \).

Step 6: Compare to the Given Options

Compare \( \frac{nM - x_n + x'}{n} \) with the given options: **(A)** \(M - x_n + x'\); **(B)** \(\frac{nM - x_n + x'}{n}\); **(C)** \(\frac{(n-1)M + x'}{m}\); **(D)** \(\frac{M - x_n + x'}{n}\). The matching formula is **(B)**.

Key Concepts

Sequence of Numbers

In mathematics, a sequence of numbers is simply an ordered list of numbers. These numbers follow a specific pattern or rule. For instance, the sequence \( x_1, x_2, x_3, \ldots, x_n \) indicates that each number in the sequence is represented by \( x \) with a subscript indicating its position in the sequence. In particular, this exercise involves \( n \) numbers, which means our sequence has exactly \( n \) terms.

The average, or the mean, of this sequence is calculated by summing all these numbers and then dividing by the number of terms, \( n \). This is given by the formula:

• \( M = \frac{x_1 + x_2 + x_3 + \cdots + x_n}{n} \).

This formula reflects how averages work by distributing the total value evenly across each term in the sequence.

New Average Calculation

Calculating a new average after changing an element in the sequence requires understanding how the previous average was affected by the sequence's elements. When one number in a sequence is replaced, it affects the total sum of the sequence, hence altering the average.

Initially, the total sum of the sequence was calculated using:

• \( nM = x_1 + x_2 + x_3 + \cdots + x_n \).

Once you replace \( x_n \) with a new number, \( x' \), you need to adjust the total sum to:

• \( nM - x_n + x' \).

The new average then becomes:

• \( \frac{nM - x_n + x'}{n} \).

This process ensures that the calculation accurately reflects the updated composition of the sequence. Calculating this new average prioritizes keeping the balance and fairness that is fundamental in the concept of a mean or average.

Replacement of a Number

Replacing a number in a sequence significantly impacts the average calculation. It is important to handle this change methodically to ensure the new average is precise.

In the given exercise, we're dealing with replacing \( x_n \) by \( x' \). This action directly influences the sum of the sequence, as the removed number \( x_n \) is no longer contributing to the sum, and the new number \( x' \) takes its place.

Here's the step-by-step approach for handling such a replacement:

• Identify the initial sum of the sequence based on the average: \( nM = x_1 + x_2 + x_3 + \cdots + x_n \).
• Subtract the value being replaced, \( x_n \), from this sum.
• Add the new number, \( x' \), to this result to find the updated total sum.
• Finally, calculate the new average using this adjusted sum: \( \frac{nM - x_n + x'}{n} \).

Understanding this process ensures that when any element of a sequence is replaced, the overall effect on the average can be accurately understood and calculated.