If `1/(b-a)+1/(b-c)=1/a+1/c` , then a,b,c are in (A) AP (B) GP (C) HP (D) NONE

Options

A. AP
B. GP
C. HP
D. None of these

Answer

C

Explanation

Suppose a, b and c are in HP.
`:." "b=(2ac)/(a+c)`
Now, consider
`(1)/(b-a)+(1)/(b-c)=(1)/((2ac)/(a+c)-a)+(1)/((2ac)/(a+c)-c)`
`=(a+c)/(a(c-a))+(a+c)/(c(a-c))=((a+c)/(c-a))((1)/(a)-(1)/(c))`
`=(a+c)/(c-a)xx(c-a)/(ca)=(a+c)/(ca)=((a+c)/(c-a))((1)/(a)-(1)/(c))`
`=(a+c)/(c-a)xx(c-a)/(ca)=(a+c)/(ca)=(1)/(a)+(1)/(c)`
Thus, our supposition is correct.
Hence, a, b and c are in HP.