Problem 11 - A Complete Resource Book in Mathematics for JEE Main 2017

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

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

Step 1: Understand the Definition of Arithmetic Mean

The A.M. (Arithmetic Mean) of a series of numbers is calculated by dividing the total sum of the numbers by the number of terms in the series. If the mean of the series is given by \( \bar{x} \) and there are \( n \) numbers in the series, then the total sum \( S \) of the series is \( n \bar{x} \).

Step 2: Identify Known Information

We know the sum of the first \((n-1)\) terms is \( k \). Let the \( n \)th term be \( x_n \). According to the problem statement, \( S = k + x_n \) because the sum of the first \((n-1)\) terms plus the \( n \)th term equals the total sum \( S \).

Step 3: Express the Total Sum with Known Values

Using the information from Step 1, we write the total sum of \( n \) terms as \( n \bar{x} = k + x_n \). This equation relates the given arithmetic mean, the sum of the first \((n-1)\) terms, and the \( n \)th term.

Step 4: Solve for the nth Term

Rearrange the equation \( n \bar{x} = k + x_n \) to solve for \( x_n \). Subtract \( k \) from both sides to isolate \( x_n \): \[ x_n = n \bar{x} - k \].

Step 5: Match with Given Options

Now, compare the expression for \( x_n \) which is \( n \bar{x} - k \) with the options given in the exercise. The correct option is (B) \( n \bar{x} - k \).

Key Concepts

n-th term calculation

In any sequence, the n-th term represents the specific position in a series that you wish to calculate or determine. The n-th term is crucial as it can define properties of your sequence like progression or regularity. To find the n-th term, you often use the data from the sequence like other known terms or other important values like the total sum of series (S) or the arithmetic mean (\(\bar{x}\)).

In this exercise, we are dealing with an arithmetic series where each number contributes to the overall mean. The n-th term here is identified as \(x_n\). Using the solution steps given, you can express \(x_n\) in terms of known quantities:

• Arithmetic mean \(\bar{x}\)
• Total number of terms \(n\)
• Sum of the first \((n-1)\) terms \(k\)

The key formula we use is derived from these relationships: \( n\bar{x} = k + x_n \). Solving for \(x_n\), we get \(x_n = n\bar{x} - k \), which helps calculate the value at the hand-picked term in the sequence not otherwise explicitly stated.

sum of series

The sum of a series adds all its different sequential values into a single total. This total is important in many statistical and mathematical calculations because it reflects the entire value across the sequence.

In mathematical sequences and especially in arithmetic sequences, each number contributes equally to this sum, though the specific value can vary greatly. The entire sum of our series of \(n\) numbers is \(n \bar{x}\), explained by the fact that the arithmetic mean is the sum divided by the number of terms.

In our example, we've broken the sum into two components:

• Sum of the first \((n-1)\) terms: \(k\)
• The n-th term itself: \(x_n\)

Therefore, the equation expressing this relationship is: \( n\bar{x} = k + x_n \). This concise representation enables us to both understand and calculate the sum of the series when given partial sums or series mean.

average formula

The average formula, or arithmetic mean, is a fundamental concept in statistics and mathematics. It's a way of expressing the central or typical value of a series of numbers. It's crucial for understanding how a series behaves at a typical point.

The arithmetic mean \(\bar{x}\) is calculated simply by dividing the total sum of the numbers by the count of the numbers: \(\bar{x} = \frac{S}{n}\), where \(S\) is the total sum and \(n\) is the number of terms.

In our problem, knowing the mean \(\bar{x}\) allows us to restructure and break down the sequence into manageable parts. Here, it directly relates to the total number of terms \(n\) and is essential for solving for the unknown \(n\)-th term alongside other given data.

This concept helps you understand a sequence's distribution of values and is used to interpret data more strategically, helping solve problems efficiently, such as the provided one that focuses on determining the unknown element in an arithmetic series through its average.