During an ongoing research I dealt with the concept of orders of group members. The following question remained a gap in my analysis. Any insight is appreciated.
Let $ \bar{o}(G) $ be the average order of the elements of $ G $, where $ G $ is a finite group. Let $ A_n $ be the alternating group of degree $ n $. Then \begin{align*} & \bar{o}(A_3)\simeq 2.33, \quad \quad \bar{o}(A_4)\simeq 2.58, \quad \quad \bar{o}(A_5)\simeq 3.51, \quad\quad \bar{o}(A_6)\simeq 3.91, \\ & \bar{o}(A_7)\simeq 5.00, \quad \quad \bar{o}(A_8)\simeq 6.79, \quad \quad \bar{o}(A_9)\simeq 8.35, \quad \quad \bar{o}(A_{10})\simeq 9.98, \\ & \bar{o}(A_{11})\simeq 11.18, \quad \bar{o}(A_{12})\simeq 12.41, \quad \bar{o}(A_{13})\simeq 14.86, \quad \bar{o}(A_{14})\simeq 19.17, \\ & \bar{o}(A_{15})\simeq 24.04, \quad \bar{o}(A_{16})\simeq 27.97, \quad \bar{o}(A_{17})\simeq 29.11, \quad \bar{o}(A_{18})\simeq 32.01, \\ & \bar{o}(A_{19})\simeq 37.56, \quad \bar{o}(A_{20})\simeq 43.47, \quad \bar{o}(A_{21})\simeq 54.14, \quad \bar{o}(A_{22})\simeq 63.11, \\ & \bar{o}(A_{23})\simeq 66.35, \quad \bar{o}(A_{24})\simeq 70.22, \quad \bar{o}(A_{25})\simeq 81.97. \end{align*}
My question:
If $n \geqslant 13$, show that $\bar{o}(A_{n}) > n+1$
