If abc are in GP and xy respectively arithmetic means between ab and bc then prove that1x 1y =2b

Answers

Question :

If a,b,c, are in G.P. and x,y respectively arithmetic means between a,b and b,c then prove that1/x +1/y =2/b
Submitted on 10/7/2024 | Answered by Vandana Rana

Answer :

a,b,c are in G.P(given)
b²=ac
x is A.M. of a,b (given)
x=(a+b)/2
1/x=2/(a+b).......(1)
y is AM.of b ,c (given)
y=(b+c)/2
1/y =2/(b+c).........(2)
Adding (1) and (2)
1/x+1/y =2/(a+b) +2/(b+c)

=2[(b+c+a+b)/(ab+ac+b²+bc)]
=2[(2b+c+a)/(2b²+ab +bc)]
=2/b[(2b+c+a)/(2b+c+a)]
=2/b