Find the average of first 30 multiples of 7.

Options

A. 94.8
B. 99.6
C. 104.4
D. 108.5

Answer

To find the average of the first 30 multiples of 7, we can follow these steps:

### Step 1: Identify the first multiple and the last multiple
The first multiple of 7 is:
\[ 7 \times 1 = 7 \]

The last multiple of 7 for the first 30 multiples is:
\[ 7 \times 30 = 210 \]

### Step 2: List the first 30 multiples of 7
The first 30 multiples of 7 are:
\[ 7, 14, 21, 28, 35, \ldots, 210 \]

### Step 3: Recognize that these multiples form an arithmetic progression (AP)
In this case, the first term \( a = 7 \) and the common difference \( d = 7 \).

### Step 4: Use the formula for the average of an arithmetic progression
The average \( A \) of an arithmetic progression can be calculated using the formula:
\[ A = \frac{\text{First term} + \text{Last term}}{2} \]

### Step 5: Substitute the values into the formula
Substituting the first term and the last term:
\[ A = \frac{7 + 210}{2} \]

### Step 6: Calculate the average
Now, calculate the average:
\[ A = \frac{217}{2} = 108.5 \]

### Conclusion
The average of the first 30 multiples of 7 is:
\[ \boxed{108.5} \]

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