A, B, C are in GP and M,N,O are in AP, then (O-N)log A + (M-O)log B + (N-M)log C is equal to ?

Options : 

A) (O-N)log ABC 

B) 0 

C) 1 

D) none

Answer is : B

Solution :

Given :-
1. A, B, C are in GP. So B² =  A C
2. M,N,O are in AP. So
    => O-N = N-M = x (assume),
then   => (O–M) = 2(O–N) = 2x. 

What To Find:- 
value of (O-N)log A + (M-O)log B + (N-M)log C

Procedure :-
Remind that (O-M) = 2x,so  (M-O) = -2x
Now, find value
  (O-N)log A + (M-O)log B + (N-M)log C   

=>  X log  A - 2X log B   + X log C   

=>  X (log A - 2 log B + log C)

=>  X (log A -  log B² + log C)

=>  X(log AC – log  B²)

=>  X(log B² – log  B²)

=>  X(0)

=>  0.