Problem: Let . Which is larger or ? (there may be more than one case)
Extra: Can a generlization be made about the bases?
Ok this hasn't been thought out well but I thought I would write down my initial thoughts.
For an n you have 1 set of (99 + 100) and 1 set of 101 on its own (I can't explian whats going on in my head for this part!).
The difference between the two is 98 so my guess is the change from to would happen at n=49.
Why I cant fully explain but something along the lines of that there's two things n acts on in on the LHS and one on the RHS and the RHS will 'catch' up to the LHS by 2 each time........
Ill think some more about what I'm actually trying to say but for now I think the change happens at n=49.
Umm... To further my ramblings...
Want to check if .
i.e. that
.
Clearly and , is converging towards 1 while is monotonically increasing.
Hence at some point will exceed for some value of . Just remains to find this as up until this point .
Pretty sure this is 49 but I don't know how to provide a proper proof!