Understanding the Problem: Average of Consecutive Odd Numbers

The question asks for the sum of the smallest and the largest number among a sequence of eight consecutive odd numbers whose average is given as 28.

Consecutive odd numbers are odd numbers that follow each other in order, with a difference of 2 between any two consecutive terms (e.g., 1, 3, 5, 7...).

Property of Averages in Arithmetic Progression

A set of consecutive odd numbers forms an arithmetic progression because the difference between consecutive terms is constant (which is 2). A key property of an arithmetic progression is that its average is equal to the average of the smallest and largest term.

The formula for the average of an arithmetic progression is:

\(\text{Average} = \frac{\text{Smallest Term} + \text{Largest Term}}{2}\)

Calculating the Sum of Smallest and Largest Number

We are given that the average of the eight consecutive odd numbers is 28.

Using the property mentioned above, we can write:

\(28 = \frac{\text{Smallest Number} + \text{Largest Number}}{2}\)

To find the sum of the smallest and the largest number, we can multiply both sides of the equation by 2:

\(\text{Smallest Number} + \text{Largest Number} = 28 \times 2\)

\(\text{Smallest Number} + \text{Largest Number} = 56\)

Therefore, the sum of the smallest and the largest number is 56.

Alternatively: Finding the Numbers First

For a set of an even number of consecutive terms in an arithmetic progression, the average lies exactly between the two middle terms. In this case, there are eight numbers, so the middle terms are the 4th and 5th terms.

The average is 28. Since the numbers are consecutive odd numbers, the 4th and 5th numbers must be odd numbers closest to 28.

• The odd number just below 28 is 27. This is the 4th term.
• The odd number just above 28 is 29. This is the 5th term.

Now we can list the eight consecutive odd numbers starting from the 4th term (27) and 5th term (29), moving outwards:

1. 4th term is 27.
2. 5th term is 29.
3. 3rd term is \(27 - 2 = 25\).
4. 6th term is \(29 + 2 = 31\).
5. 2nd term is \(25 - 2 = 23\).
6. 7th term is \(31 + 2 = 33\).
7. 1st term is \(23 - 2 = 21\). (Smallest number)
8. 8th term is \(33 + 2 = 35\). (Largest number)

The eight consecutive odd numbers are 21, 23, 25, 27, 29, 31, 33, and 35.

The smallest number is 21.

The largest number is 35.

The sum of the smallest and the largest number is \(21 + 35 = 56\).

Conclusion

Both methods confirm that the sum of the smallest and the largest of the eight consecutive odd numbers is 56.

Revision Table: Consecutive Number Averages

Additional Information: Finding Terms

If you know the average and the number of terms in an arithmetic progression, you can find the terms. For 'n' terms:

• If 'n' is odd, the average is the middle term ((\(n+1\)/2)th term). You can then find other terms by adding or subtracting the common difference.
• If 'n' is even, the average is the mean of the two middle terms (\((n/2)\)th and \((n/2 + 1)\)th terms). The two middle terms are equidistant from the average. The common difference for consecutive odd/even numbers is 2.