@KlondikeBob199@Math_files I think you didn't understand the problem.
I have shown an example of the difference between countable and uncountable infinity.
Look carefully at the equation of the name I have mentioned. You find there will be at least one person who is not getting a room.
@Math_files Proving above
Room 1 ABAAB.....
Room 2 BAABB....
Room 3 ABBAA....
Room 4 BABBA.....
Room 5 ABBAB....
....
Now creating a new name in the form
Li≠RiLi
Then the new will be BBAAA....
Now notice this name is not present in any of the room. As a result Everyone will not get a room.
@Math_files The given info is good but there is a limitation.
Consider a bus arrived with infinitely many passengers whose names are infinitely long with combinations of A and B. In that case you cannot provide room to everyone. This is the diff between countable and uncountable infinity.
@sonukg4india f(x)=(x-1)/(x+1)
f(f(x))=((x-1)/(x+1)-1)/((x-1)/(x+1)+1)
=(x-1-x-1)/(x-1+x+1)
=-1/x
f(f(f(f(x))))=x
Hence it is a periodic sequence
Given
a⁵⁰⁷ × (a-1)⁵⁰⁷/(a+1)⁵⁰⁷ × (-1/a)⁵⁰⁶ × (a+1)⁵⁰⁶/(1-a)⁵⁰⁶=-1
a(a-1)/(a+1)=-1
a²-a=-a-1
a²=-1
No value of a satisfy the above.