Problem:
What is the value of
1+2+3−4+5+6+7−8+⋯+197+198+199−200?
Answer Choices:
A. 9800
B. 9,900
C. 10,000
D. 10,100
E. 10,200
Solution:
Looking at the numbers, you see that every set of 4 has 3 positive numbers and 1 negative number. Calculating the sum of the first couple of sets gives us
2+10+18…+394
Clearly, this pattern is an arithmetic sequence. By using the formula we get
22+394⋅50=(B) 9900.
OR
Note that the original expression is equal to
(1+2+3+⋯+199+200)−2(4(1+2+3+⋯+49+50))
Since the sum of the first n positive integers is 2n(n+1), this is equal to
2200(201)−2(4(250(51)))
which can be simplified as
20100−4(50)(51)=20100−10200=(B) 9900.
The problems on this page are the property of the MAA's American Mathematics Competitions