Three distinct positive integers are in arithmetic progression with integer common difference d ≥ 1. The sum of the first and the third terms is 24, and all three terms are less than 15. If the product of the three numbers is maximized, what is the value of the common difference d?
Solution
Let us assume the numbers to be a-d, a and a+d.
Sum of 1st and 3rd terms => a-d + a+d = 24
==> a = 12.
As all terms are less than 15, 12+d < 15 and d < 3.
Possible values for d = 1 or 2.
If d = 1, terms become 11, 12, 13 and their product = 1716.
If d = 2, terms become 10, 12, 14 and their product = 1680.
So, to maximize the product, value of difference should be 1.
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