Calculating the Nth Number Using Arithmetic Mean

This solution explains how to find the value of the nth number when the arithmetic mean of \(n\) numbers is given, along with the sum of the first \((n-1)\) numbers.

Understanding Arithmetic Mean

The arithmetic mean (or average) of a set of numbers is calculated by dividing the sum of all the numbers by the count of the numbers in the set.

Mathematically, if we have \(n\) numbers, let them be \(x_1, x_2, ..., x_n\). The arithmetic mean (\(M\)) is given by the formula:

\( M = \frac{x_1 + x_2 + ... + x_n}{n} = \frac{\sum_{i=1}^{n} x_i}{n} \)

Applying the Formula to the Problem

We are given:

• The total count of numbers is \(n\).
• The arithmetic mean of these \(n\) numbers is \(M\).
• The sum of the first \((n - 1)\) numbers is \(k\). Let this sum be \(S_{n-1} = k\).
• We need to find the \(n\)th number, let's call it \(x_n\).

Step-by-Step Solution

1. Calculate the total sum of the \(n\) numbers: From the definition of the arithmetic mean, we know that:

   \( M = \frac{\text{Sum of } n \text{ numbers}}{n} \)

   Rearranging this formula to find the sum of all \(n\) numbers (let's call it \(S_n\)):

   \( S_n = M \times n \)

   So, the total sum of the \(n\) numbers is \(nM\).

2. Relate the total sum to the given sum: The total sum (\(S_n\)) is the sum of the first \((n - 1)\) numbers plus the \(n\)th number (\(x_n\)).

   \( S_n = (x_1 + x_2 + ... + x_{n-1}) + x_n \)

   We know that the sum of the first \((n - 1)\) numbers is \(k\). So:

   \( S_n = k + x_n \)

3. Solve for the \(n\)th number (\(x_n\)): Now we have two expressions for the total sum \(S_n\):
   • \(S_n = nM\)
   • \(S_n = k + x_n\)

   By setting these two expressions equal to each other:

   \( nM = k + x_n \)

   To find the \(n\)th number (\(x_n\)), we isolate it by subtracting \(k\) from both sides of the equation:

   \( x_n = nM - k \)

Conclusion

Therefore, the value of the nth number is \(nM - k\). This is derived directly from the definition of the arithmetic mean and the information provided about the sum of the first \((n-1)\) terms.