mikegonz00
New member
- Joined
- Mar 20, 2021
- Messages
- 11
Hello,
I'm reading Calculus: Deconstructed and came across the sequence xi=(xi-1+1)/xi-2, i ≥ 2. After letting x0=a and x1=b and calculating the next five terms of the sequence, I found that x5=a and x6=b, which implies that the sequence is periodic since it's a second-order recurrence: xi = xi+5k, i,k ≥ 0.
At first glance, I didn't anticipate the sequence to be periodic. My question is: Is there a method to generate periodic sequences defined recursively? I could think of an inefficient way to check whether the sequence is periodic (basically the method I used above), but what properties would we want to include in our recurrence relation when trying to create our own periodic sequence?
I'm reading Calculus: Deconstructed and came across the sequence xi=(xi-1+1)/xi-2, i ≥ 2. After letting x0=a and x1=b and calculating the next five terms of the sequence, I found that x5=a and x6=b, which implies that the sequence is periodic since it's a second-order recurrence: xi = xi+5k, i,k ≥ 0.
At first glance, I didn't anticipate the sequence to be periodic. My question is: Is there a method to generate periodic sequences defined recursively? I could think of an inefficient way to check whether the sequence is periodic (basically the method I used above), but what properties would we want to include in our recurrence relation when trying to create our own periodic sequence?