7. For how many positive integers, ,is ?

 

 

We want to know how many integers are between 250 and 500, not including either of those numbers.  By looking for a pattern, we can simplify the problem.

 

Between 250 and 251 there are no integers:  251 – 250 = 1

Between 250 and 252 there is one integer (251):  252 – 250 = 2

Between 250 and 253 there are two integers (251 and 252):  253 – 250 = 3

 

It appears the answer we are looking for is one less than the difference between the numbers:  (500 – 250) – 1 = 250 – 1 = 249.