Understanding Consecutive Odd Numbers and Averages

This problem asks us to compare the average of the first five consecutive odd numbers with the average obtained when the next three consecutive odd numbers are included. Let's break it down step by step.

Identifying the First Five Consecutive Odd Numbers

Consecutive odd numbers follow each other in sequence, with a difference of 2 between any two adjacent numbers. The first odd number is 1.

So, the first five consecutive odd numbers are:

• 1
• 3
• 5
• 7
• 9

Calculating the Average of the First Five Odd Numbers (m)

The average of a set of numbers is the sum of the numbers divided by the count of the numbers.

Sum of the first five odd numbers = $1 + 3 + 5 + 7 + 9 = 25$

Number of values = 5

Average (m) = $\frac{\text{Sum}}{\text{Count}} = \frac{25}{5} = 5$

So, the initial average, m, is 5.

Identifying the Next Three Consecutive Odd Numbers

The last number in our initial set is 9. The next odd number after 9 is $9 + 2 = 11$.

The next three consecutive odd numbers after 9 are:

• 11
• 13 ($11 + 2$)
• 15 ($13 + 2$)

Calculating the Average Including the Next Three Odd Numbers

Now, we include these three numbers with the first five. The new set of numbers is the first eight consecutive odd numbers:

1, 3, 5, 7, 9, 11, 13, 15

Sum of the first eight odd numbers = (Sum of first five) + (Sum of next three)

Sum of first eight odd numbers = $25 + (11 + 13 + 15) = 25 + 39 = 64$

Number of values = 8

New Average = $\frac{\text{Sum}}{\text{Count}} = \frac{64}{8} = 8$

Finding the Increase in the Average

The increase in the average is the difference between the new average and the original average (m).

Increase in average = New Average - Original Average (m)

Increase in average = $8 - 5 = 3$

Therefore, the increase in the average is 3.

Revision Table: Average of Consecutive Numbers

Here's a quick summary of the concepts used:

• Consecutive Odd Numbers: Numbers that follow each other in order, with a difference of 2 (e.g., 1, 3, 5, ...).
• Average: The sum of a set of values divided by the number of values. It represents a central value of the set.

Additional Information: Properties of Averages

When dealing with arithmetic sequences (like consecutive odd numbers), the average is often the middle term if the number of terms is odd. For an even number of terms, it's the average of the two middle terms.

• For 1, 3, 5, 7, 9 (5 terms), the middle term is 5, which is the average.
• For 1, 3, 5, 7, 9, 11, 13, 15 (8 terms), the middle terms are 7 and 9. The average is $\frac{7+9}{2} = \frac{16}{2} = 8$. This matches our calculation.

Adding consecutive numbers to an arithmetic sequence tends to shift the average towards the numbers being added. Since we added larger odd numbers (11, 13, 15), the average increased.