A subsemigroup $K$ of $S$ is said to be monogenic if $K = \langle a \rangle$ for some $a \in S$ and the order of $a$ is the size of $K$. If the order of $a$ is finite, then after some time power of $a$ will be repeated. Then the set $\{ x \in \mathbb N : (\exists y \in \mathbb N ) a^x = a^y, x \neq y \}$ is non empty and so it has a least element says $m$ is called the index of $a$. The least element $r \in \mathbb N$ satisfy $a^{m+ r} = a^m$ is called the period of $a$. Every finnite monogenic subsemigroup can be represented by $M(m, r) = \langle a \rangle$. I have found that the index of $a^i \in \langle a \rangle$, where $i < m$ is either $q$ or $q + 1$, where $ m = iq + r$ by divison algorithm theorem.
I want to find out what is the period of $a^i$, where $i < m$ ?
I would be thankful for any kind of help.