What is the difference between the sum of all even integers

Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

What is the difference between the sum of all even integers between 51 and 100 (inclusive) and the sum of all even integers between 1 and 50 (inclusive)?

A. 1,250
B. 2,000
C. 2,500
D. 3,750
E. 5,000

To find the sum of a set of numbers, we first find the arithmetic average of the set and then multiply by the number of items in the set.

The sum of the even integers from 51 to 100, which is really from 52 to 100, is:

sum = average x number of items = (100 + 52)/2 x [(100 - 52)/2 + 1]

sum = 76 x 25 = 1,900

The sum of the even integers from 1 to 50, which is really from 2 to 50 is:

sum = average x number of items = (2 + 50)/2 x [(50 - 2)/2 + 1]

26 x 25 = 650

Thus, the difference is 1,900 - 650 = 1,250.

Alternate solution:

(52 + 54 + … + 98 + 100) - (2 + 4 + … + 48 + 50)

= (52 - 2) + (54 - 4) + … + (98 - 48) + (100 - 50)

= 50 + 50 + … + 50 + 50 (Note: Since there are there are twenty-five pairings, there are twenty-five 50s.)

= 50 x 25

= 1,250

Answer: A