+0  
 
0
309
1
avatar+1442 

If $a$ and $b$ are integers with $a > b$, what is the smallest possible positive value of $\frac{a+b}{a-b} + \frac{a-b}{a+b}$?

AnonymousConfusedGuy  Oct 13, 2017
 #1
avatar+87301 
+2

\($\frac{a+b}{a-b} + \frac{a-b}{a+b}$\)

 

[ (a + b)(a + b) + (a - b)(a - b) ]  / [ a^2 - b^2 ] =

 

 ( [ a^2 + 2ab + b^2  + a^2 - 2ab +b^2 ] / [ a^2 - b^2]  =

 

[ 2a^2 + 2b^2 ] / [ a^2 - b^2 ]

 

2 [ a^2 + b^2] / [ a^2 - b^2 ]

 

The smallest positive value, 2, is achieved when  b  = 0  and a is any positive integer

 

In every other case,  [ a^2 + b^2] / [ a^2 - b^2 ] will be > 1   and the expression will be > 2

 

 

cool cool cool

CPhill  Oct 13, 2017
edited by CPhill  Oct 13, 2017

9 Online Users

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.