Understanding Consecutive Even Numbers

Consecutive even numbers are even integers that follow each other in sequence. For example, 4, 6, 8 are consecutive even numbers. The difference between any two adjacent numbers in such a sequence is always 2.

A key property of consecutive numbers (including consecutive even numbers) is that their average is equal to the median. When there is an even number of terms in the sequence (like six numbers here), the average falls exactly halfway between the two middle terms.

Finding the Six Consecutive Even Numbers

We are given that the average of six consecutive even numbers is 13.

Let the six consecutive even numbers be represented as $n_1, n_2, n_3, n_4, n_5, n_6$.

Since the average is 13 and there are six numbers, the average must lie between the 3rd ($n_3$) and 4th ($n_4$) numbers.

Mathematically, this can be written as:

Multiplying both sides by 2, we get the sum of the two middle numbers:

Since $n_3$ and $n_4$ are consecutive even numbers, $n_4$ must be 2 greater than $n_3$. So, we can write $n_4 = n_3 + 2$.

Now, substitute this into the sum equation:

Combine like terms:

Subtract 2 from both sides:

Divide by 2 to find $n_3$:

Now we can find $n_4$:

So, the two middle numbers are 12 and 14. Working backwards and forwards, we can find the entire sequence of six consecutive even numbers:

• The number before 12 is $12 - 2 = 10$.
• The number before 10 is $10 - 2 = 8$.
• The number after 14 is $14 + 2 = 16$.
• The number after 16 is $16 + 2 = 18$.

Therefore, the six consecutive even numbers are 8, 10, 12, 14, 16, and 18.

Calculating the Next Three Even Numbers

The last number in our sequence of six consecutive even numbers is 18. We need to find the average of the *next three* consecutive even numbers.

These numbers will follow 18 in the sequence of even numbers:

• The first next even number is $18 + 2 = 20$.
• The second next even number is $20 + 2 = 22$.
• The third next even number is $22 + 2 = 24$.

So, the next three consecutive even numbers are 20, 22, and 24.

Determining the Average of the Next Three Numbers

To find the average of these three numbers (20, 22, 24), we sum them and divide by the count (which is 3).

Sum = $ 20 + 22 + 24 $

Sum = 66

Average = $ \frac{\text{Sum}}{\text{Count}} $

Average = $ \frac{66}{3} $

Average = 22

Thus, the average of the next three even numbers is 22.