Answer:
The sixth number is 22. Surprisingly, the exact values of the first five numbers do not matter. See the explanation.
Step-by-step explanation:
Let the sixth number be [tex]x[/tex]. The average of [tex]n[/tex] numbers is equal to the sum of these number divided by [tex]n[/tex]. That is:
[tex]\displaystyle \text{Average} = \frac{\text{Sum of Entries}}{\text{Numbers of Entries}}[/tex].
Working backwards (multiply both sides by the denominator, [tex]n[/tex]) to find a formula for the sum of the numbers:
[tex]\text{Sum of Entries} = n \cdot \text{Average of Entries}[/tex].
Apply this formula to find the sum of the first five numbers:
[tex]\underbrace{4}_{\text{Avg. of}\atop\text{entries}} \times \underbrace{5}_{\text{Num. of}\atop \text{entries}} = 20[/tex].
If [tex]x[/tex] represents the value of the sixth number, the sum of the first six numbers will be equal to:
[tex]20 + x[/tex].
The average of the first six number will thus be equal to:
[tex]\displaystyle \text{Average of the first six numbers} = \frac{\text{Sum of Entries}}{\text{Numbers of Entries}}= \frac{20 + x}{6}[/tex].
However, the question states that the average of the first six numbers is equal to [tex]7[/tex]. In other words,
[tex]\displaystyle \frac{20 + x}{6} = 7[/tex].
Multiply both sides by the denominator, [tex]6[/tex]:
[tex]20 + x = 42[/tex].
[tex]x = 22[/tex].
In other words, the sixth number is [tex]22[/tex].
