Final answer:
The inequality (nr) < (nr+1) holds when 0 ≤ r < ½(n -1), which ensures that the addition of 1 to r significantly increases the product nr for n ≥ 1.
Step-by-step explanation:
To prove that for n ≥ 1: (nr) < (nr+1) if and only if 0 ≤ r < ½(n -1), we can start by understanding the relationship between n, r, and their products. For any positive n, increasing r by 1 will definitely increase the product nr. However, the inequality in question is specific about the relation between n and r for this to be true.
We are given that r must be at least 0 and less than half of (n - 1), which ensures that when we add 1 to r, the increase is significant enough to make the inequality hold. If r is equal to or larger than ½(n - 1), then the addition of 1 to r could make (nr+1) less than or equal to (nr), especially when n is small.