Let A be a set of distinct positive integers. If the arithmetic mean (average) of the elements of A is 25, what is the maximum possible value of an element in A?
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Expert's answer
2012-08-14T09:58:37-0400
Suppose A consists of n distinct positive integers
a_1,a_2,...,a_n. Since their mean is 25 we have that
a_1+a_2+...+a_n=25*n
We can assume that the numbers are ordered, and so a_1 < a_2 < ... < a_n.
Hence the maximum possible value for an will be achieved, when
a_1=1 a_2=2 ... a_{n-1} = n-1
By formula for the sum of arithmetic progression we have
1+2+...+n-1 = (n-1)(n-2)/2,
Therefore
a_n = 25*n - (1+2+...+n-1) = 25 n - (n-1)(n-2)/2 = 25 n - (n^2 + n +2n +2 ) /2 = 25 n - n^2/2 + 1.5n + 1 = - n^2/2 + 26.5n + 1
We should find maximum of the function g(n) = - n^2/2 + 26.5n + 1 among all integer positive numbers.
Let us find critical points of g, ie. solutions of the equation g'(n)=0.
We have that g'(n) = -n + 26.5 = 0
whence n = 26.5
Hence the maximum of g anomg positive integers is achieved either at n=26 or at n=27.
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