What is the average of the first 20 multiples of 3?

To find the average of the first 20 multiples of 3, we can follow a straightforward process. First, we need to identify what those multiples are, then calculate their sum, and finally divide that sum by the number of multiples to find the average.

Identifying the First 20 Multiples of 3

The first 20 multiples of 3 can be generated by multiplying 3 by each of the first 20 natural numbers. Here’s how they look:

• 3 x 1 = 3
• 3 x 2 = 6
• 3 x 3 = 9
• 3 x 4 = 12
• 3 x 5 = 15
• 3 x 6 = 18
• 3 x 7 = 21
• 3 x 8 = 24
• 3 x 9 = 27
• 3 x 10 = 30
• 3 x 11 = 33
• 3 x 12 = 36
• 3 x 13 = 39
• 3 x 14 = 42
• 3 x 15 = 45
• 3 x 16 = 48
• 3 x 17 = 51
• 3 x 18 = 54
• 3 x 19 = 57
• 3 x 20 = 60

Calculating the Sum

Next, we need to sum these multiples. Instead of adding each one individually, we can use a formula for the sum of an arithmetic series. The sum \( S \) of the first \( n \) multiples of a number \( a \) can be calculated using the formula:

S = n/2 * (first term + last term)

In our case, \( n = 20 \), \( a = 3 \), the first term is 3, and the last term is 60. Plugging these values into the formula gives:

S = 20/2 * (3 + 60) = 10 * 63 = 630

Finding the Average

Now that we have the sum of the first 20 multiples of 3, we can find the average by dividing the sum by the number of terms:

Average = Sum / Number of Terms

Average = 630 / 20 = 31.5

Final Result

The average of the first 20 multiples of 3 is 31.5. This method not only provides the answer but also illustrates how to approach similar problems involving averages and multiples in mathematics.