Each of A and B is a positive real number and N is an integer with N > 1 satisfying:
AN - A - 1 = 0, and:
B2N - B – 3A = 0
Which of A and B is greater?
It is relatively easy to show that both A and B decrease when N grows bigger.
Looking at both equations one can see that the above values are never less than 1, each of them approach 1 as a limit, and we want to see which is bigger for the same N.
Assume A=k*B
And denote 2N by t
WLOG B=1+b
Value of b getting smaller as t increases
Since B^t-B=3k*B
and B^t>1+bT
1+tb-1-b=b*(t-1)
And for any N > 2 t-1>3
k>b*(t-1)/(3*b) = (t-1)/3 >1 ergo A>B
as suspected
Edited on August 29, 2016, 1:46 am
more than suspicion ... spoiler
