Problem:
The product (1−221​)(1−321​)⋯(1−921​)(1−1021​) equals
Answer Choices:
A. 125​
B. 21​
C. 2011​
D. 32​
E. 107​
Solution:
Factor each term of the given expression as the difference of two squares and group the terms according to signs to obtain
[(1−21​)(1−31​)(1−41​)⋯(1−101​)][(1+21​)(1+31​)(1+41​)⋯(1+101​)]=[21​32​43​⋯109​][23​34​45​⋯1011​]=[101​][211​]=2011​​
OR
Note that
(1−221​)=43​(1−221​)(1−321​)=43​⋅98​=32​=64​(1−221​)(1−321​)(1−421​)=64​⋅1615​=85​.​
Clearly the pattern is
(1−221​)⋯(1−n21​)=2nn+1​.
(This may be proved by induction.) Thus the answer is 2011​.