Question

If $a-b,b-c,c-a$ are in G.P. then find the relation among $a,b$ and $c$

Answer 1

$a-b,b-c,c-a$ are in G.P

$\Rightarrow \frac{b-c}{a-b}=\frac{c-a}{b-c}$

$\Rightarrow {(b-c)}^2=(a-b)(c-a)$

$\Rightarrow b^2+c^2-2bc=ac-a^2-bc+ab$

$\Rightarrow a^2+b^2+c^2-ab-bc-ac=0$

$\Rightarrow \frac{1}{2}(2a^2+2b^2+2c^2-2ab-2bc-2ac)=0$

$\Rightarrow \frac{1}{2}(a^2-2ab+b^2+b^2-2bc+c^2+c^2-2ac+a^2)=0$

$\Rightarrow {(a-b)}^2+{(b-c)}^2+{(c-a)}^2=0$

$\Rightarrow a-b=0,b-c=0,c-a=0$

$\Rightarrow a=b,b=c,c=a$

$\therefore a=b=c$ is the relation among $a,b$ and $c$