Problem:
A cart rolls down a hill, traveling 5 inches the first second and accelerating so that during each successive 1-second time interval, it travels 7 inches more than during the previous 1-second interval. The cart takes 30 seconds to reach the bottom of the hill. How far, in inches, does it travel?
Answer Choices:
A. 215
B. 360
C. 2992
D. 3195
E. 3242
Solution:
The distance traveled during the 1st, 2 nd, 3rd,…,30 th second is 5,5+7,5+2⋅7,…,5+29⋅7, respectively. Therefore the total distance traveled is
5+(5+7)+(5+2⋅7)+⋯+(5+29⋅7)
=5⋅30+(7+2⋅7+⋯+29⋅7)
=150+7⋅(1+2+⋯+29)
=150+7⋅229⋅30​
= (D) 3195​.
The problems on this page are the property of the MAA's American Mathematics Competitions