The average of the first 18 positive multiples of 15 is:

The correct option is 142.5.

Step-by-step Explanation:

We are asked to find the average of the first 18 positive multiples of 15.

The first 18 positive multiples of 15 are:

$15\times 1$

$15\times 2$

$15\times 3$

...

$15\times 18$

Method 1: Using the formula for the sum of an Arithmetic Progression (AP)

These multiples form an arithmetic progression where:

First term (a) = 15

Last term (l) = 15 × 18 = 270

Total number of terms (n) = 18

The formula for the average of an arithmetic progression is given by:

$\text{Average}=\frac{\text{First term}+\text{Last term}}{2}$

Substituting the values into the formula:

$\text{Average}=\frac{15+270}{2}$

$\text{Average}=\frac{285}{2}$

$\text{Average}=142.5$

Method 2: Using the sum of first n natural numbers

The sum of the first 18 multiples of 15 can be written as:

$\text{Sum}=15\times (1+2+3+...+18)$

The sum of the first n natural numbers is given by the formula:

$S_n=\frac{n(n+1)}{2}$

For n = 18:

$\text{Sum of natural numbers}=\frac{18\times 19}{2}=9\times 19=171$

Now, total sum of the multiples:

$\text{Total Sum}=15\times 171=2565$

The average is the total sum divided by the number of terms (18):

$\text{Average}=\frac{2565}{18}=142.5$

Thus, the average of the first 18 positive multiples of 15 is 142.5.