Doing Sequences in a cool way

Try to solve it without looking downward

1.)

One can think of it till the cows come home or use the formula, but there is another cool way:
check it out:



Obvously the answer will be one over three, one third.

there is another picture:


So there is another question:


2.)

But I can't think of one in a geometrical way.
If you can, please tell me.


and there is another question:


3.)

what is the picture?

pretty obviously, the answer becomes one over eight

ok, let us think of another question:


4.)
I have no idea for the geometrical picture. Or is there such a geometrical picture?

So what is the trick?

Why some of could them be easily found, but some else we hardly had any idea?

what is in commen between 1.) and 3.) ? and what else is in common between 2.) and 4.)?