(d)

We have,

$C_1, C_2, C_3, \ldots, C_n$ be circle with radii

$r_1, r_2, \ldots, r_n$ respectively. $C_i$ and $C_{i+1}$ touch externally $X$-axis and $y=2 \sqrt{2} x+10$ are tangent of each circle.

Slope of line $y=2 \sqrt{2} x+10$ is $2 \sqrt{2}$

$\therefore \quad \tan 2 \theta =2 \sqrt{2}$

$\frac{2 \tan \theta}{1-\tan ^2 \theta} =2 \sqrt{2}$

$\sqrt{2} \tan ^2 \theta+\tan \theta-\sqrt{2}=0$

$(\sqrt{2} \tan \theta-1)(\tan \theta+\sqrt{2}) =0$

$\tan \theta=\frac{1}{\sqrt{2}} \tan \theta \neq-\sqrt{2}$

$\sin \theta=\frac{1}{\sqrt{3}}$

$\operatorname{In} \Delta P Q M, \sin \theta=\frac{Q M}{P Q}$

$\Rightarrow \quad P Q=\sqrt{3} Q M \Rightarrow P Q=\sqrt{3} r_1$

$\operatorname{In} \triangle P R N$,

$\sin \theta=\frac{R N}{P R}=\frac{r_2}{P Q+r_1+r_2}=\frac{r_2}{\sqrt{3} r_1+r_1+r_2}$

$\Rightarrow \quad \frac{1}{\sqrt{3}}=\frac{r_2}{(\sqrt{3}+1) r_1+r_2}$

$\Rightarrow r_2+(\sqrt{3}+1) r_1=\sqrt{3}-r_2$

$\Rightarrow \quad \frac{r_2}{r_1}=\frac{\sqrt{3}+1}{\sqrt{3}-1}$

$r_1, r_2, r_3$ a geometric progression with common ratio $2+\sqrt{3}$.