Answer: 2
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Step-by-step explanation:
Let x be the smallest even integer. The next one up is x+2. Then after that is (x+2)+2 = x+4, etc. Each time we need the next even integer, we add on 2.
We have this list:
- x = first
- x+2 = second
- x+4 = third
- x+6 = fourth
- x+8 = fifth
We'll add up those expressions to get
(x)+(x+2)+(x+4)+(x+6)+(x+8)
(x+x+x+x+x)+(2+4+6+8)
5x+20
Then we divide that over 5 since we want to find the average. Set that result equal to the desired average of 6 and solve for x.
(5x+20)/5 = 6
5x+20 = 5*6
5x+20 = 30
5x = 30-20
5x = 10
x = 10/5
x = 2 is the smallest number in the list
The rest of the list is:
- x+2 = 2+2 = 4
- x+4 = 2+4 = 6
- x+6 = 2+6 = 8
- x+8 = 2+8 = 10
The full list being: {2,4,6,8,10}
Adding up everything in the list gets us: 2+4+6+8+10 = 30
Divide that over five to find the average: 30/5 = 6
We get the proper arithmetic mean, so this confirms the correct answer.
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Here's another approach:
Start with the average 6. This is the center of the list of five values. Because it's the center, we spread out by going 2 units up and 2 units down from this center. So we'll go from {6} to {4,6,8}. This expansion process is then applied again to go from {4,6,8} to {2,4,6,8,10}. At this point, we can see there are five values here and the center is definitely 6. More importantly, the smallest item here is 2