Problem:
The infinite product
310⋅3310⋅33310⋯
evaluates to a real number. What is that number?
Answer Choices:
A. 10
B. 3100
C. 41000
D. 10
E. 10310
Solution:
Observe that
310⋅3310⋅33310⋯=1031⋅1091⋅10271⋯=1031+91+271+⋯=1031⋅1−311=1021=(A)10
OR
Let
x=310⋅3310⋅33310⋯
Cubing both sides of the equation yields
x3=10⋅310⋅3310⋯=10x
Because each factor is greater than 1 , the product cannot equal 0 or be negative, so x=(A)10.
The problems on this page are the property of the MAA's American Mathematics Competitions