The sum of the positive odd integers less than 50 is subtracted from the sum of the positive even integers less than or equal to 50. What is the resulting difference? a) 0

Difficulty:

Hard b) 25 c) 50 d) 100 e) 200 HINT To solve this problem, you do

not

have to find the value of the sum of the positive odd integers less than 50 nor the value of the sum of the positive even integers less than or equal to 50. When you subtract the first sum from the second sum, the terms can be regrouped in a way that makes the computation much easier.

B: 25

Explanation

The sum of the positive odd integers less than 50 is

1 + 3 + 5 + ··· + 45 + 47 + 49

, and the sum of the positive even integers less than or equal to 50 is

2 + 4 + 6 + ··· + 46 + 48 + 50

. When the first sum is subtracted from the second, the terms can be regrouped to form

(2 – 1) + (4 – 3) + ··· + (48 – 47) + (50 – 49)

, which is just

1 + 1 + 1 + ··· + 1

, where there are 25 ones. Therefore, the correct answer is 25.