I need to prove that if a sequence $\{a_n\}_{n\in\Bbb N}$ is such that $$ a_n = \frac{a_{n-1}+a_{n+1}}{2} \quad\forall n\in\Bbb N $$ then the sequence is arithmetic progression.
I transformed that formula to the following one: $${a_n-a_{n-1}}=a_{n+1}-a_n$$ And at this point it's pretty obvious that we could say the diffrence between the element and his neighbors is the same, so it is an arithmetic progression.
But I think for a strong proof I need to use induction, but i don't really understand, how could I solve it with induction. I know how to used, but how to apply it here?
Thanks!