The mean of the values of 1, 2, 3, ... x with respective frequencies y, 2y, 3y, ..., xy is

Options

A. (2x+1)
B. x(x+1)(2x+1)
C. `(2x+1)/(3)`
D. None of these

Answer

To find the mean of the values \(1, 2, 3, \ldots, x\) with respective frequencies \(y, 2y, 3y, \ldots, xy\), we will follow these steps:

### Step 1: Identify the Observations and Frequencies
The observations are \(x_i = 1, 2, 3, \ldots, x\) and the corresponding frequencies are \(f_i = y, 2y, 3y, \ldots, xy\).

### Step 2: Calculate the Total Frequency \(N\)
The total frequency \(N\) is given by the sum of all frequencies:
\[ N = f_1 + f_2 + f_3 + \ldots + f_x = y + 2y + 3y + \ldots + xy \]
This can be factored as:
\[ N = y(1 + 2 + 3 + \ldots + x) = y \cdot \frac{x(x + 1)}{2} \]
(using the formula for the sum of the first \(n\) natural numbers).

### Step 3: Calculate the Weighted Sum of the Observations
Next, we calculate the weighted sum of the observations \( \sum x_i f_i \):
\[ \sum x_i f_i = 1 \cdot y + 2 \cdot 2y + 3 \cdot 3y + \ldots + x \cdot xy \]
This can be expressed as:
\[ \sum x_i f_i = y(1^2 + 2^2 + 3^2 + \ldots + x^2) = y \cdot \frac{x(x + 1)(2x + 1)}{6} \]
(using the formula for the sum of the squares of the first \(n\) natural numbers).

### Step 4: Calculate the Mean
The mean \( \bar{x} \) is given by the formula:
\[ \bar{x} = \frac{\sum x_i f_i}{N} \]
Substituting the values we calculated:
\[ \bar{x} = \frac{y \cdot \frac{x(x + 1)(2x + 1)}{6}}{y \cdot \frac{x(x + 1)}{2}} \]
The \(y\) and \(\frac{x(x + 1)}{2}\) terms cancel out:
\[ \bar{x} = \frac{\frac{x(x + 1)(2x + 1)}{6}}{\frac{x(x + 1)}{2}} = \frac{2x + 1}{3} \]

### Final Answer
Thus, the mean of the values \(1, 2, 3, \ldots, x\) with respective frequencies \(y, 2y, 3y, \ldots, xy\) is:
\[ \bar{x} = \frac{2x + 1}{3} \]