The sum of the first 80 positive odd integers subtracted from the sum of the first 80 positive even integers is:
(a) 0 (b) 20 (c) 40 (d) 60 (e) 80
Solution
Here's a nice method:
E = 2 + 4 + 6 + 8 + 10 + ... + 158 + 160
O = 1 + 3 + 5 + 7 + 9 + ... + 157 + 159
We want:
E - O = (2-1) + (4-3) + (6-5) + (8-7) + (10-9) + ... + (158-157) + (160-159)
= 1 + 1 + 1 + 1 + ... + 1 + 1
= 80.
Here's another method:
The sum of an arithmetic series is n/2 \cdot (f - l) where there are n terms, f is the first, and l is the last.
So E = 80/2 \cdot (2 + 160) = 6480, and O = 80/2 \cdot (1 + 159) = 6400.
Again, the difference is 80. So the answer is (e).