Explanation

(c)
Since, $a,\ b,\ c$ are in AP.
$\therefore\ 2b = a + c$
$\Rightarrow \ 2bx = (a + c)x\ for\ all\ x$
$\Rightarrow \ 2bx = (a + c)x + 20\ for\ all\ x$
$\Rightarrow \ 2(bx + 10) = (ax + 10) + (cx + 10)$
$\therefore\ \ 10^{2(bx + 10)} = 10^{(ax + 10) + (cx + 10)}$
$\Rightarrow \ \left( 10^{bx + 10} \right)^{2} = 10^{ax + 10}.10^{cx + 10}$
$\Rightarrow \ 10^{ax + 10},\ 10^{bx + 10},\ 10^{cx + 10}$
Are in GP for all $x$