Consider a decreasing sequence of positive integers that satisfies the following conditions:
What is the greatest possible value of
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From the second condition, we have that the average of the first terms is , the average of the first terms is ... so inductively the average of the first terms is . This also implies that the sum of the first terms is .
If we know the partial sums of a sequence, we may recover the terms by subtracting two adjacent terms. In other words, the th term of this sequence is the sum of the first terms minus the sum of the first terms. This gives us:
As the terms must be positive: gives us the maximum possible value of as
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Additionally, note that since , it is possible to assign values to , , and which does not affect the bound later down the line.
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