How to prove the result of orbiting of a symmetric relation is also symmetric?

Question

Suppose $R$ is a symmetric relation on a set $A$. I want to prove $R^n$(i.e $R\circ R\circ\dots\circ R$) is also symmetric. I approach the problem by considering first the odd numbers, for instance $n=3$. $$R\circ R\circ R = R(R(R(x))) = R(R(y)) = R(x) = y$$ So $xRy$ analoge $yRx$. Then I consider the even numbers: $n=2$ $$R\circ R = R(R(x)) = R(y) = x $$ So I proved for even numbers that the orbiting results in reflexive relation.

How can I now prove the rest?

Answer 1

Assume aRR...RRb. Thus some u,v,... y,z with
aRu, uRv,... yRz, zRb. By symmetry bRz, zRy,... vRu, uRa.
Consequently bRR...RRa, QED.

What does the functional notation R(x) mean?