Let
![R_1,R_2, \ldots](/media/m/c/7/3/c736401f25f145971d67c4a60f4b0f94.png)
be the family of finite sequences of positive integers defined by the following rules:
![R_1 = (1),](/media/m/f/c/5/fc5c815f94d6169e0e4cc6fdc5ce86b3.png)
and if
![R_{n - 1} = (x_1, \ldots, x_s),](/media/m/b/7/8/b787c6c3f96f82e784899e0466b5cc6c.png)
then
![R_n = (1, 2, \ldots, x_1, 1, 2, \ldots, x_2, \ldots, 1, 2, \ldots, x_s, n).](/media/m/9/8/b/98bb89067b7693abd3d0daf6131551dd.png)
For example,
![R_4 = (1, 1, 1, 2, 1, 2, 3, 4).](/media/m/f/1/6/f165a6899a374ab6648a28c84d9599bd.png)
Prove that if
![n > 1,](/media/m/a/5/5/a5594d91e9be511e99e3bbed777bbe12.png)
then the
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
th term from the left in
![R_n](/media/m/d/0/e/d0e65e2d0606ef2f0de3ff9955d825b9.png)
is equal to 1 if and only if the
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
th term from the right in
![R_n](/media/m/d/0/e/d0e65e2d0606ef2f0de3ff9955d825b9.png)
is different from 1.
%V0
Let $R_1,R_2, \ldots$ be the family of finite sequences of positive integers defined by the following rules: $R_1 = (1),$ and if $R_{n - 1} = (x_1, \ldots, x_s),$ then
$$R_n = (1, 2, \ldots, x_1, 1, 2, \ldots, x_2, \ldots, 1, 2, \ldots, x_s, n).$$
For example, $R_2 = (1, 2),$ $R_3 = (1, 1, 2, 3),$ $R_4 = (1, 1, 1, 2, 1, 2, 3, 4).$ Prove that if $n > 1,$ then the $k$th term from the left in $R_n$ is equal to 1 if and only if the $k$th term from the right in $R_n$ is different from 1.