Problem:
Suppose that (un​) is a sequence of real numbers satisfying un+2​=2un+1​+un​, and that u3​=9 and u6​=128. What is u5​?
Answer Choices:
A. 40
B. 53
C. 68
D. 88
E. 104
Solution:
Note that u5​=2u4​+9 and 128=u6​=2u5​+u4​=5u4​+18. Thus u4​=22, and it follows that u5​=2⋅22+9=(B)53​.
The problems on this page are the property of the MAA's American Mathematics Competitions