Let us take series $a^2$. Where $a = 1, 2, 3, …$
The entire series of $a^2$ is look like: $A = 1, 4, 9, 16, …$
First step: The difference between every two terms of $A$ is: $B$ = $3, 5, 7, …$
Second step: The difference between $B$ is: $2, 2,… $ and looks like arithmetic progression.
In this problem, we have taken $a^2$ and within second step we have seen common difference and concluded this series as arithmetic series.
If you take $a^3$, within $3rd$ step we can conclude common difference is $3! = 6$ and arithmetic series.
My question is: How to prove the series $a^n$ for $a = 1, 2, 3,…$ gives common difference n! at $n$ th step, and gives arithmetic series. Also, I want to know any such similar observations are made so far.