Let $s_1,\ldots,s_k $ are elements of $G$. Let $\langle s_1,\ldots, s_k \rangle$ be the set of elements generated by $\{s_1,\ldots,s_k\}$. Let $s_{k+1} \notin \langle s_1,\ldots, s_k \rangle$. Then it is a well known fact that
$$\langle s_1,\ldots,s_k \rangle \le 2\langle s_1,\ldots,s_{k+1} \rangle$$
The above hold in case of a group. Let $G$ be a group without associativity(loop). Is this still true? In other words do we need associativity to prove the above inequality?