Problem:
n a sequence of positive integers, each term after the second is the product of the previous two terms. The sixth term in the sequence is . What is the first term?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Let be the first term and be the second. Then we get the following sequence:
From this, we get that . Factoring , we get . We need and to be integers. The only fifth powers that divide are and . If , then , which doesn't work since is not a perfect cube. Therefore, . This forces to equal .
Thus, is the correct answer.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions