Prove: (Choose only one) If a > b > 0 then a^n > b^n for every positive integer n or If a > b then a > a+b/2 > bUse the typical “statement - reason” method with axiom,...

Prove: (Choose only one) If a > b > 0 then a^n > b^n for every positive integer n or If a > b then a > a+b/2 > b

Use the typical “statement - reason” method with axiom, definition, or theorem in proving.

Expert Answers
beckden eNotes educator| Certified Educator

The answer is if a>b>0 then a^n > b^n

Proof by induction

a)  a>b ==> ````a^1 > b^1  by definiton.
b) suppose a^(n-1) > b^(n-1)
c) We know that if a>b>0 and c>d>0 then a*c > b*d
and by our supposition a*a^(n-1) > b*b^(n-1) and this gives
a^n > b^n
So we have proved that if a>b>0 then a^n > b^n by induction.