Why doesn't mathematical induction work backwards or with increments other than $1$? From my understanding of my topic, if a statement is true for $n=1$, and you assume a statement is true for arbitrary integer $k$ and show that the statement is also true for $k+1,$ then you prove that the statement's true for all $n\geq 1$. Makes sense.
However - why can't I do this backwards? If I show the statement is true for $k-1,$ aren't I showing that if the statement is true for $n=1,$ it's likewise true for $n=0,n=-1,n=-2,\ldots$?
Also, why can't I prove the statement is true for $k+0.1$, and prove the statement true for $n=1.1,1.2,1.3,\ldots$? Both of these scenarios, in my mind, seem to follow the same logic as the "proper" definition of mathematical induction - but apparently they're no-go. Can someone please explain why?
Thank you!
Edit: The consensus seems to be that yes, even though it's abnormal, induction as I've stated above it is logically sound. Which raises the question - why has my math teacher said this is wrong? Is it as I suspect, where she didn't want me straying from the proper definition of $k+1$ induction and possibly confusing myself (or losing points on the test), or is there something else that makes the above fundamentally flawed?
Thanks!

This kind of induction has both it goes backwards and has increments other than 1.
