Problem:
For all integers , the value of
is always which of the following?
Answer Choices:
A. a multiple of 4
B. a multiple of 10
C. a prime number
D. a perfect square
E. a perfect cube
Solution:
We first expand the expression:
\dfrac{(n+2)(n+1) n!-(n+1) n!}
We can now divide out a common factor of ! from each term of the numerator:
Factoring out , we get
which proves that the answer is
OR
In the numerator, we factor out an ! to get
Now, without loss of generality, test values of until only one answer choice is left valid:
, knocking out , and . , knocking out . This leaves a perfect square as the only answer choice left.
This solution does not consider the condition . The reason is that, with further testing it becomes clear that for all , we get
as proved in Solution . The condition was added most likely to encourage picking and discourage substituting smaller values into .
The problems on this page are the property of the MAA's American Mathematics Competitions