Average Property of Consecutive Numbers Explained

This problem asks us to find the largest number within a set of 7 consecutive numbers, given that their average is 20. Consecutive numbers follow each other in order, with a difference of 1 between each number (e.g., 3, 4, 5, 6).

Understanding the Average of Consecutive Numbers

A useful property of consecutive numbers is that their average is always the middle number, provided there is an odd count of numbers in the sequence. Since we have 7 numbers, which is an odd count, the average (20) corresponds directly to the middle number of the sequence.

For a sequence of 7 numbers, the middle number is the $\frac{7 + 1}{2} = 4$th number.

Therefore, the 4th number in our sequence is 20.

Determining the Sequence

Knowing that the 4th number is 20, we can list the sequence:

• 4th number: 20

To find the numbers following the 4th number, we add 1 successively:

• 5th number: $20 + 1 = 21$
• 6th number: $21 + 1 = 22$
• 7th number (the largest): $22 + 1 = 23$

To find the numbers preceding the 4th number, we subtract 1 successively:

• 3rd number: $20 - 1 = 19$
• 2nd number: $19 - 1 = 18$
• 1st number: $18 - 1 = 17$

The complete sequence of 7 consecutive numbers is 17, 18, 19, 20, 21, 22, 23.

We can check the average: $\frac{17 + 18 + 19 + 20 + 21 + 22 + 23}{7} = \frac{140}{7} = 20$. This confirms our sequence is correct.

Identifying the Largest Number

The question asks for the largest of these numbers. In the sequence 17, 18, 19, 20, 21, 22, 23, the largest number is the last one.

The largest number is 23.