Question

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

• A. A.P.
• B. G.P.
• C. H.P.
• D. none of these

Answer 1

The correct option is A A.P.

Let:

$v_1=a+\frac{1}{bc}$

$v_2=b+\frac{1}{ac}$

$v_3=c+\frac{1}{ab}$

Given:

$2b=a+c$ [$a,b,c$ are in A.P.]

Let us check if $v_1,v_2,v_3$ are in A.P.:

$v_1+v_3=a+\frac{1}{bc}+c+\frac{1}{ab}$

$v_1+v_3=a+c+\frac{1}{b}\left(\frac{a+c}{ac}\right)$

Substituting $a+c=2b$:

$v_1+v_3=2b+\frac{1}{b}\left(\frac{2b}{ac}\right)$

$v_1+v_3=2b+\left(\frac{2}{ac}\right)$

$v_1+v_3=2(b+\left(\frac{1}{ac}\right))$

$v_1+v_3=2v_2$

$\therefore$ $v_1,v_2,v_3$ are in A.P.