Think of the sequence of numbers...what type of sequence must they be?
Hey there I need some help with this proof.
There are 2000 points on a circle, and each point is given a number that is equal to the average of its two nearest neighbors. Show that all the numbers must be equal.
I am not sure how to start this. I hope someone can guide me step by step on how to do this proof, so that I can understand this better.
I'm not exactly sure what this sequence is called but this pattern is that the term your on is the average of the 2 closest neighbors, so it is the previous term plus the next term divided by 2?
Sorry if I'm way off, I take a longer time to learn than others
Let the sequence of numbers be denoted by
We require:
where and
This results in the homogeneous recursion:
whose characteristic equation is:
hence:
and so we see we must have an arithmetic sequence. We then may write:
and so we have:
Now, we also know we must have:
hence:
and so we have: