if x,y,z are in arithmetic progression,then prove that (1) 1÷yz,1÷zx,1÷xy are in A.P
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we have to prove that 1÷(yz)-1÷(Z-Y= 1/(xy)-1/(zx)
X,y,z are in A.P
.
. . Y-X=Z-Y...........(1)
Now,
LHS
1/(zx)-1/(Z-Y)
Z(Y-X)/(z²xy)
(Y-X)/zxy
RHS
1/(xy)-1/(zx)
x(z-y)/x²zy
(z-y)/xyz
LHS=RHS. [Y-X=Z-Y ( From EQ( 1)]
X,y,z are in A.P
.
. . Y-X=Z-Y...........(1)
Now,
LHS
1/(zx)-1/(Z-Y)
Z(Y-X)/(z²xy)
(Y-X)/zxy
RHS
1/(xy)-1/(zx)
x(z-y)/x²zy
(z-y)/xyz
LHS=RHS. [Y-X=Z-Y ( From EQ( 1)]
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