Problem:
The first four terms in an arithmetic sequence are x+y,x−y,xy, and x/y, in that order. What is the fifth term?
Answer Choices:
A. −815
B. −56
C. 0
D. 2027
E. 40123
Solution:
Since the difference of the first two terms is −2y, the third and fourth terms of the sequence must be x−3y and x−5y. Thus
x−3y=xy and x−5y=yx
so xy−5y2=x. Combining these equations we obtain
(x−3y)−5y2=x and, therefore, −3y−5y2=0
Since y cannot be 0, we have y=−3/5, and it follows that x=−9/8. The fifth term in the sequence is x−7y=123/40.
Answer: E.
The problems on this page are the property of the MAA's American Mathematics Competitions