Problem:
Suppose that the number a satisfies the equation 4=a+a−1. What is the value of a4+a−4?
Answer Choices:
A. 164
B. 172
C. 192
D. 194
E. 212
Solution:
Squaring each side of the equation 4=a+a−1 gives
16=a2+2a⋅a−1+(a−1)2=a2+2+a−2, so 14=a2+a−2
Squaring again gives
196=a4+2a2⋅a−2+(a−2)2=a4+2+a−4, so (D)194=a4+a−4
The problems on this page are the property of the MAA's American Mathematics Competitions