Problem:
Given that
​x1​=211x2​=375x3​=420x4​=523, and xn​=xn−1​−xn−2​+xn−3​−xn−4​ when n≥5​
find the value of x531​+x753​+x975​.
Solution:
For n>5,
xn​​=xn−1​−xn−2​+xn−3​−xn−4​=(xn−2​−xn−3​+xn−4​−xn−5​)−xn−2​+xn−3​−xn−4​=−xn−5​​
It follows that the sequence repeats in a cycle ten terms long. Hence
x531​+x753​+x975​​=x1​+x3​+x5​=x1​+x3​+x4​−x3​+x2​−x1​=x4​+x2​=523+375=898​.​
OR
Using the theory of difference equations, a characteristic equation for the sequence is x4=x3−x2+x−1 or x4−x3+x2−x+1=0. Since x5+1=(x+1)(x4−x3+x2−x+1), we can conclude xn​+xn−5​=0 and proceed as above.
The problems on this page are the property of the MAA's American Mathematics Competitions