Mathematics - Sequence & Series Question with Solution | TestHub

$225a^2+9b^2+25c^2-75ac-45ab-15bc=0$

$\Rightarrow 2\left({\left(15a\right)}^2+{\left(3b\right)}^2+{\left(5c\right)}^2-\left(15a\right)\left(5c\right)-\left(15a\right)\left(3b\right)-\left(3b\right)\left(5c\right)=0 \right)$

$\Rightarrow 2{\left(15a\right)}^2+2{\left(3b\right)}^2+2{\left(5c\right)}^2-2\left(15a\right)\left(5c\right)-2\left(15a\right)\left(3b\right)-2\left(3b\right)\left(5c\right)=0$

$\Rightarrow {\left(15a-3b\right)}^2+{\left(3b-5c\right)}^2+{\left(5c-15a\right)}^2=0$

$\Rightarrow 15a-3b=3b-5c=5c-15a=0$

$\Rightarrow 15a=3b=5c$

$\Rightarrow \frac{a}{1}=\frac{b}{5}=\frac{c}{3}=k$

$\Rightarrow a=k, b=5k, c=3k$

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

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