SOLUTION: We may define the three means (arithmetic, geometric, and harmonic) of two positive numbers a and b as A = (a+b)/2 G = √ab H = 2ab/ a+b, respectively c. show that 1

Algebra ->  Finance -> SOLUTION: We may define the three means (arithmetic, geometric, and harmonic) of two positive numbers a and b as A = (a+b)/2 G = √ab H = 2ab/ a+b, respectively c. show that 1      Log On


   

Question 1120747: We may define the three means (arithmetic, geometric, and harmonic) of two positive numbers a and b as
A = (a+b)/2
G = √ab
H = 2ab/ a+b, respectively
c. show that 1/ H-a + 1/ H-b = 1/a + 1/b
Answer by amarjeeth123(570) About Me  (Show Source):
You can put this solution on YOUR website!
A = (a+b)/2
G = √ab
H = 2ab/ a+b
H-a=(2ab/a+b)-a=(2ab-a^2-ab/a+b)=(ab-a^2/a+b)
H-b=(2ab/a+b)-b=(2ab-ab-b^2)/(a+b)=(ab-b^2/a+b)
1/H-a=a+b/ab-a^2
1/H-b=a+b/ab-b^2
1/H-a+1/H-b=a+b(1/a(b-a)+1/b(a-b))=a+b(1/b(a-b)-1/a(a-b))=a+b(a-b/ab(a-b))=a+b/ab=1/a+1/b
Hence proved.