Respuesta :
Answer: S+0.5(n+1)
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Explanation:
Consider the set
[tex]\left\{x_1,x_2,x_3,\ldots,x_n\right\}\\\\[/tex]
which has n items in it. We don't know anything about any individual item, but what we do know is that the mean is S.
So,
[tex]\text{mean} = \frac{\text{sum of values}}{\text{number of values}}\\\\\text{mean} = \frac{x_1+x_2+x_3+\ldots+x_n}{n}\\\\S = \frac{x_1+x_2+x_3+\ldots+x_n}{n}\\\\nS = x_1+x_2+x_3+\ldots+x_n\\\\x_1+x_2+x_3+\ldots+x_n = nS\\\\[/tex]
In short, the sum of the values [tex]\left\{x_1,x_2,x_3,\ldots,x_n\right\}\\\\[/tex] is exactly nS, where S is the mean of the same set.
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Next, we add 1 to [tex]x_1[/tex], add 2 onto [tex]x_2[/tex], etc to form this new set
[tex]\left\{x_1+1,x_2+2,x_3+3,\ldots,x_n+n\right\}\\\\[/tex]
Let's find the mean like so
[tex]\text{mean} = \frac{\text{sum of values}}{\text{number of values}}\\\\\text{mean} = \frac{(x_1+1)+(x_2+2)+(x_3+3)+\ldots+(x_n+n)}{n}\\\\\text{mean} = \frac{\left(x_1+x_2+x_3+\ldots+x_n\right)+\left(1+2+3+\ldots n\right)}{n}\\\\\text{mean} = \frac{nS+0.5n(n+1)}{n}\\\\\text{mean} = \frac{n(S+0.5(n+1))}{n}\\\\\text{mean} = S+0.5(n+1)\\\\[/tex]
which is why S+0.5(n+1) is the answer
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Let's look at a concrete example.
Consider the set {1,7,10} which has n = 3 items in it.
Its mean is,
S = (1+7+10)/3 = 18/3 = 6
Now increase the items in {1,7,10} by adding on the values {1,2,3} to each element in the order shown
1+1 = 2
7+2 = 9
10+3 = 13
So {1,7,10} becomes {2,9,13}
The mean of the new set {2,9,13} is...
mean = (2+9+13)/3 = 24/3 = 8
Notice how,
newMean = S + 0.5(n+1)
newMean = 6 + 0.5(3+1)
newMean = 8
which helps us see how the formula works and partially confirm the answer. While this example isn't definitive proof, it's still handy to go over. I recommend forming other sets to check to see that this formula works. Not only do I recommend trying different values, but also a different number of values as well (i.e. increase n to some larger integer).
