Problem 11 - A Complete Resource Book in Mathematics for JEE Main

Question

The A.M. of \(n\) numbers of a series is \(\bar{x}\). If the sum of the first \((n-1)\) term is \(k\), them the \(n\)th number is (A) \(\bar{x}-k\) (B) \(n \bar{x}-k\) (B) \(\bar{x}-n k\) (D) \(n \bar{x}-n k\)

Step-by-Step Solution

Answer

(B) \(n \bar{x} - k\)

Step 1: Understand the Given Information

We have a series of \(n\) numbers where the arithmetic mean (A.M.) is \(\bar{x}\). This means the total sum of these \(n\) numbers is given by \(n \bar{x}\). We are informed that the sum of the first \((n-1)\) terms in this series is \(k\).

Step 2: Express the Total Sum of the Series

Since the arithmetic mean is \(\bar{x}\), the total sum of the \(n\) numbers is given by: \(n \bar{x}\).

Step 3: Determine the n-th Term

The n-th term, which we can denote as \(x_n\), can be found by subtracting the sum of the first \((n-1)\) terms from the total sum of \(n\) terms. Thus, \(x_n = n \bar{x} - k\).

Step 4: Identify the Correct Option

Compare the expression for the n-th term, found in Step 3, with the given options. Option (B) is \(n \bar{x} - k\), which matches our derived expression.

Key Concepts

Series Sum

In mathematics, understanding the concept of a series sum is fundamental when dealing with sequences of numbers. The series sum is essentially the total of a sequence of numbers added together. Often, these numbers can form specific patterns, like arithmetic sequences.

When you are given a series with an arithmetic mean, like in our problem, the series sum can be easily calculated. If you know the arithmetic mean \( \bar{x} \) of \( n \) numbers, the total sum of these numbers is simply \( n \cdot \bar{x} \). This is because the arithmetic mean is calculated by dividing the total sum of the series by the number of items. Hence, multiplying the mean by the number of terms gives the total sum.

In situations where you have additional pieces of information, such as the sum of a subset of terms in the series, it helps in solving for unknown terms or details about the series.

Arithmetic Sequence

An arithmetic sequence is a type of sequence where each term is derived from the previous one by adding a constant value, known as the common difference. For instance, in a simple arithmetic sequence like 2, 4, 6, 8, the common difference is 2. Each number is obtained by adding 2 to the previous number.

When you are dealing with an arithmetic sequence, it can be useful to know the general formula to find any term in the sequence. The nth term \( a_n \) of an arithmetic sequence can be calculated using:

• \( a_n = a_1 + (n-1) \, d \)

where \( a_1 \) is the first term and \( d \) is the common difference. Understanding this pattern makes solving problems with sequences much simpler.

In the exercise, although it is not explicitly a problem about finding sequence terms through a common difference, recognizing the summation pattern and structure helps in deriving solutions.

n-th Term Calculation

The n-th term calculation is a crucial part of understanding the broader scope of series and sequences. Given the total sum of a series and the sum of its parts, calculating the n-th term involves simple arithmetic operations.

In the provided problem, you are tasked with finding the n-th term given that the arithmetic mean of the entire series is known and the sum of the first \((n-1)\) terms of the series is provided. To find this n-th term, you simply subtract the known sum of the first \((n-1)\) terms from the total series sum.

This is expressed mathematically as:

• \( x_n = n \bar{x} - k \)

where \( x_n \) is the n-th term, \( n \bar{x} \) represents the total sum of the series, and \( k \) is the sum of the first \((n-1)\) terms.

Such steps make the calculation straightforward by applying a clear, logical approach to uncover the value of the unknown term.