The variance of first 10 multiples of 3 is

Options

A. 64.25
B. 54.25
C. 70.25
D. 74.25

Answer

To find the variance of the first 10 multiples of 3, we will follow these steps:

### Step 1: Identify the first 10 multiples of 3
The first 10 multiples of 3 are:
\[ 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 \]

### Step 2: Calculate the sum of these multiples
We can use the formula for the sum of an arithmetic series:
\[ S_n = \frac{n}{2} \times (a + l) \]
where:
- \( n \) = number of terms = 10
- \( a \) = first term = 3
- \( l \) = last term = 30

Calculating:
\[ S_{10} = \frac{10}{2} \times (3 + 30) = 5 \times 33 = 165 \]

### Step 3: Calculate the mean
The mean (\( \bar{x} \)) is given by:
\[ \bar{x} = \frac{\sum x_i}{n} \]
Substituting the values:
\[ \bar{x} = \frac{165}{10} = 16.5 \]

### Step 4: Calculate the sum of squares of the multiples
We need to calculate:
\[ \sum x_i^2 = 3^2 + 6^2 + 9^2 + 12^2 + 15^2 + 18^2 + 21^2 + 24^2 + 27^2 + 30^2 \]
Calculating each square:
\[ = 9 + 36 + 81 + 144 + 225 + 324 + 441 + 576 + 729 + 900 \]
Adding these values:
\[ = 9 + 36 + 81 + 144 + 225 + 324 + 441 + 576 + 729 + 900 = 4035 \]

### Step 5: Calculate the variance
The formula for variance (\( \sigma^2 \)) is:
\[ \sigma^2 = \frac{\sum x_i^2}{n} - \bar{x}^2 \]
Substituting the values:
\[ \sigma^2 = \frac{4035}{10} - (16.5)^2 \]
Calculating:
\[ = 403.5 - 272.25 = 131.25 \]

### Final Answer
The variance of the first 10 multiples of 3 is:
\[ \boxed{131.25} \]