Question

If log a,log b,log c are in A.P. and also log a-log 2b,log 2b-log 3c,log 3c-log a are in A.P., then

• A. a,b,c are in H.P.
• B. a,2b,3c are in A.P.
• C. a,b,c are the sides of a triangle
• D. none of the above

Hint:

If logarithms are in A.P., the numbers form a G.P.

Answer 1

The Correct Option is C

Step 1: From log a,log b,log c in A.P.: 2log b=log a+log c ⟹ b²=ac
Step 2: Second A.P. condition gives: (c)/(b)=(2)/(3) ⟹ a=(3)/(2)b
Step 3: Hence a:b:c=9:6:4, which satisfy triangle inequalities.