Question

If $a,b,c$ are in A.P. then show that $\frac{1}{bc},\frac{1}{ca},\frac{1}{ab}$ will be in A.P.

Answer 1

Since $a,b,c$ are in A.P. then we have $b-a=c-b$.....(1).

We are to prove that $\frac{1}{bc},\frac{1}{ca},\frac{1}{ab}$ are also in A.P.

Now,

$\frac{1}{ca}-\frac{1}{bc}$

$=\frac{(b-a)}{abc}$

$=\frac{(c-b)}{abc}$ [Using (1)]

$=\frac{1}{ab}-\frac{1}{ca}$

$\frac{1}{ca}-\frac{1}{bc}$$=\frac{1}{ab}-\frac{1}{ca}$

$\Rightarrow$$\frac{1}{bc},\frac{1}{ca},\frac{1}{ab}$ are in A.P.