Question

If a, b, c are in A.P., then the following are also in A.P.
$a(\frac{1}{b}+\frac{1}{c}),b(\frac{1}{c}+\frac{1}{a}),c(\frac{1}{a}+\frac{1}{b})$.

• A. True
• B. False

Answer 1

The correct option is A True

We have,

$a,b,c$ are in $A.P$

$\frac{a}{abc},\frac{b}{abc},\frac{c}{abc}$ are in $A.P$

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

$\frac{ab+bc+ca}{bc},\frac{ab+bc+ca}{ca},\frac{ab+bc+ca}{ab}$ are also in $A.P$

$\frac{ab+bc+ca}{bc}-1,\frac{ab+bc+ca}{ca}-1,\frac{ab+bc+ca}{ab}-1$ are also in $A.P$

$\frac{ab+ca}{bc},\frac{ab+bc}{ca},\frac{bc+ca}{ab}$ are also in $A.P$

$\frac{a(b+c)}{bc},\frac{b(c+a)}{ca},\frac{c(a+b)}{ab}$ are also in $A.P$

$a(\frac{1}{b}+\frac{1}{c}),b(\frac{1}{c}+\frac{1}{a}),c(\frac{1}{a}+\frac{1}{b})$ are also in $A.P$

Hence, this is the answer.