Question

Let a, b, and c be distinct real numbers which are in G.P. If x $\in$ R is such that a+ x, b+x, and c+x are in H.P., then x equals

• A. a
• B. b
• C. c
• D. (a+b+c)/3

Answer 1

The correct option is B

b

As a, b, c are distinct and are in G.P., so a, b, c cannot be in A.P. Hence

$2b\ne a+c$

Now,$\frac{1}{a+x},\frac{1}{b+x},\frac{1}{c+x}$ are in A.P. Hence,

$\Rightarrow (a-b)(c+x)=(a+x)(b-c)$

$\Rightarrow (ac-bc-ab+ac)=(2b-a-c)x$

$\Rightarrow b(2b-a-c)=(2b-a-c)x[\because ac=b^2]$

$\Rightarrow x=b$