Randy Marsh

If there's 'no scope' for $(1+1)$ then you can't "do it infinitely".

@GigiBayte2 There has to be an array as input or some other structure that records order. Otherwise we're not talking about a binary search. If your input is the integer interval $[1,n]$, then what is exactly the motivation of doing a binary search, since you can detect whether $i\in\mathbb N$ is in it in constant time?

You aren't saying anything about the input, so there can't be a formula. The sorted input arrays $[1,1,1,1,5]$ and $[1,5,5,5,5]$ will need a different number of probes for the same $s$, $f$ and $n$.

There are many professional mathematicians with advanced degrees on this website and it's not particularly helpful for your cause to disrespectfully call them "still students". If mathematics could be reduced to metaphors, we wouldn't be bothering with rigor and every textbook would be like Alice in Wonderland. To paraphrase Santayana, if you can't remember anything, you won't learn anything.

I'm not sure what you mean by "pictorial way". The Cantor Pairing Function is a "pictorial" proof of the countability of rationals, but it's still a well-defined function. Pictures and metaphors are not always possible, some concepts are too abstract to be faithfully represented by mental images and you have to learn how to understand them in their abstract setting.

What level are you aiming at? Do you want an elementary treatment of these topics, or would you be comfortable with a textbook that would eventually introduce you to abstract algebra, e.g. groups, rings, fields, Galois theory etc?