Question

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

Answer 1

As ,b and c are in A.P
$2b=a+c$
and
$\frac{1}{ab}+ \frac{1}{bc}= \frac{b(a+c)}{ab(bc)}= \frac{a+c}{abc}= \frac{2b}{abc}= \frac{2}{ca}$
so
$\frac{1}{bc}, \frac{1}{ca}, \frac{1}{ab}$ are in A.P
then
$a+ \frac{1}{bc},b+ \frac{1}{ca},c+ \frac{1}{ab}$
are in A.P
as sum of two A.P's is in an A.P