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Which is greater? (Posted on 2016-08-28) Difficulty: 3 of 5
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?

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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Solution more than suspicion ... spoiler Comment 3 of 3 |

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
  Posted by Ady TZIDON on 2016-08-29 01:42:36

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