Problem:
Which of the following is equivalent to
(2+3)(22+32)(24+34)(28+38)(216+316)(232+332)(264+364)?
Answer Choices:
A. 3127+2127
B. 3127+2127+2â‹…363+3â‹…263
C. 3128−2128
D. 3128+2128
E. 5^
Solution:
The product telescopes as follows:
(2+3)(22+32)(24+34)(28+38)⋯(264+364)
=(3−2)(3+2)(32+22)(34+24)(38+28)⋯(364+264)
=(32−22)(32+22)(34+24)(38+28)⋯(364+264)
=(34−24)(34+24)(38+28)⋯(364+264)
â‹®
=(364−264)(364+264)
=(C)3128−2128​.
The problems on this page are the property of the MAA's American Mathematics Competitions