The arithmetic mean of \(n\) numbers is \(M\). If the sum of first \((n - 1)\) terms is \(k\), then what is the \(n\)th number?
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.
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:
\( M = \frac{x_1 + x_2 + ... + x_n}{n} = \frac{\sum_{i=1}^{n} x_i}{n} \)
We are given:
\( M = \frac{\text{Sum of } n \text{ numbers}}{n} \)
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 \(nM\).
\( S_n = (x_1 + x_2 + ... + x_{n-1}) + x_n \)
We know that the sum of the first \((n - 1)\) numbers is \(k\). So:
\( S_n = k + x_n \)
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:
\( x_n = nM - k \)
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.
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