Proof on this word problem

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.

Re: Proof on this word problem

Think of the sequence of numbers...what type of sequence must they be?

Re: Proof on this word problem

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

Re: Proof on this word problem

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:

Re: Proof on this word problem

Thank you so much for your help, I understand this a lot better now