The average of the first 18 positive multiples of 14 is:

The correct option is 133.

To find the average of the first 18 positive multiples of 14, we can break down the calculation step-by-step.

The first 18 positive multiples of 14 are:
14 × 1, 14 × 2, 14 × 3, ..., 14 × 18

This forms an arithmetic progression (AP) where the first term is 14, the common difference is 14, and the number of terms is 18.

The sum of these 18 multiples can be written as:
Sum = 14 × 1 + 14 × 2 + 14 × 3 + ... + 14 × 18

We can factor out 14 from each term:
Sum = 14 × (1 + 2 + 3 + ... + 18)

The sum of the first $n$ positive integers is given by the formula:
$S=\frac{n(n+1)}{2}$

Here, $n=18$, so the sum of integers from 1 to 18 is:
$1+2+3+...+18=\frac{18\times 19}{2}=9\times 19=171$

Now, we substitute this back into our sum equation:
Sum = 14 × 171 = 2394

The average is defined as the total sum divided by the number of terms (which is 18):
$\text{Average}=\frac{\text{Sum}}{18}=\frac{2394}{18}$

Dividing 2394 by 18 gives:
Average = 133

Alternative Method:
For any arithmetic progression, the average of the terms is simply the average of the first and the last term.
First term ($a_1$) = 14 × 1 = 14
Last term ($a_{18}$) = 14 × 18 = 252

Average is:
$\text{Average}=\frac{14+252}{2}=\frac{266}{2}=133$