Let $S = 8+88+888+ \ldots +\underbrace{888\dots8}_{n\text { times}}$
$\implies S = 8\left [1+11+111+ \ldots + \underbrace{111\dots 1}_{n\text { times}} \right]$
$\implies S = \dfrac{8}{9}\left [9+99+999+ \ldots + \underbrace{999\dots 9}_{n\text { times}}\right]$
$\implies S = \dfrac{8}{9}\left [(10-1)+(10^{2}-1)+(10^{3}-1)+ \ldots + (10^{n} – 1) \right]$
$\implies S = \dfrac{8}{9}\left [\underbrace{10^{1} + 10^{2} + 10^{3}+ \ldots + 10^{n}}_{\textbf{GP Series}}\:- n \right]$
$\implies S = \dfrac{8}{9}\left[\dfrac{10(10^{n} - 1)}{10-1} - n \right]\:\:\:\:\left[\because\text{Sum of GP}: S_{n} = \dfrac{a(r^{n} - 1)}{r-1}\:\:; r>0\:\: \text{and}\:\: S_{n} = \dfrac{a(1- r^{n})}{1-r}\:\:; r< 0 \right]$
$\implies S = \dfrac{8}{9}\left[\dfrac{10^{n+1} - 10-9n}{9}\right]$
$\implies S = \dfrac{8}{81}\left[10^{n+1} - 9n - 10 \right]$
$\implies S = \dfrac{8}{81}\left[10^{n} \cdot 10 - 9n - 10 \right]$
$\implies S = \dfrac{80}{81} (10^{n} – 1) – \dfrac{8}{9}n$
$\textbf{Shortcut:}$ We can check, by taking the values of $n = 1,2,3,\dots.$
So, the correct answer is $(D).$