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)]=[213243⋯109][233445⋯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.