Question 15
Consider the set S = {2, 3, 4, ...., 2n+1}, where n is a positive integer larger than 2007. Define X as the average of the odd integers in S and Y as the average of the even integers in S. What is the value of X - Y ?
Solution
The odd numbers in the set are 3, 5, 7, ...2n+1
Sum of the odd numbers = 3+5+7+...+(2n+1) = $$n^2 + 2n$$
Average of odd numbers = $$n^2 + 2n$$/n = n+2
Sum of even numbers = 2 + 4 + 6 + ... + 2n = 2(1+2+3+...+n) = 2*n*(n+1)/2 = n(n+1)
Average of even numbers = n(n+1)/n = n+1
So, difference between the averages of even and odd numbers = 1
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