Question

The arithmetic mean of n number series is $\bar{x}$. The sum of last n-1 numbers is k. Then the first number will be:

• A. 1
• B. $n\bar{x} - k$
• C. (n-1) $\bar{x}$ - k
• D. k

Hint:

To test this algebraic relationship, plug in simple numbers:
- Let the series be \(\{2, 4, 6\}\). Here, \(n=3\).
- Mean \(\bar{x} = (2+4+6)/3 = 4\).
- Sum of the last \(n-1\) numbers (4 and 6) is \(k = 10\).
- Calculate the first number: \(n\bar{x} - k = (3 \times 4) - 10 = 12 - 10 = 2\).
This matches our first number, confirming the formula works.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The arithmetic mean (average) is calculated by dividing the sum of a set of values by the total number of values in that set.
Using this relationship, we can determine individual values if the overall mean and partial sums are known.

Step 2: Key Formula or Approach:
The formula for the arithmetic mean \(\bar{x}\) of \(n\) numbers (\(x_1, x_2, \dots, x_n\)) is:
\[ \bar{x} = \frac{\sum_{i=1}^n x_i}{n} \] Rearranging this formula gives the total sum of all values:
\[ \sum_{i=1}^n x_i = n\bar{x} \]

Step 3: Detailed Explanation:
Let us calculate the value of the first number (\(x_1\)) step-by-step:
1. The total sum of all \(n\) numbers in the series is:
\[ S_n = x_1 + x_2 + x_3 + \dots + x_n = n\bar{x} \] 2. We are given that the sum of the last \(n-1\) numbers (from \(x_2\) to \(x_n\)) is \(k\):
\[ S_{n-1} = x_2 + x_3 + \dots + x_n = k \] 3. The total sum can be written as the sum of the first number and the sum of the remaining numbers:
\[ S_n = x_1 + S_{n-1} \] 4. Substituting the values into this equation:
\[ n\bar{x} = x_1 + k \] 5. Solving for the first number \(x_1\):
\[ x_1 = n\bar{x} - k \] Therefore, the first number is \(n\bar{x} - k\).

Step 4: Final Answer:
The first number in the series is \(n\bar{x} - k\), matching Option (B).