+0

0
422
1
+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
+90968
+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

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