Sequences and Series: Sum of Geometric Progression with Repeating Digits

$S_n = 8(1 + 11 + 111 + 1111 + \dots \text{ to } n \text{ terms})$

$S_n = \frac{8}{9} (9 + 99 + 999 + 9999 + \dots \text{ to } n \text{ terms})$

$9 = 10 - 1, \quad 99 = 10^2 - 1, \quad 999 = 10^3 - 1, \dots$ $S_n = \frac{8}{9} [(10 - 1) + (10^2 - 1) + (10^3 - 1) + \dots + (10^n - 1)]$ $= \frac{8}{9} [(10 + 10^2 + 10^3 + \dots + 10^n) - (1 + 1 + 1 + \dots + 1)]$ $= \frac{8}{9} [(10 + 10^2 + \dots + 10^n) - n]$

The series $10 + 10^2 + \dots + 10^n$ is a G.P. with first term $a = 10$, common ratio $r = 10$, and $n$ terms: $\sum_{k=1}^n 10^k = \frac{a(r^n - 1)}{r - 1} = \frac{10(10^n - 1)}{10 - 1} = \frac{10(10^n - 1)}{9}$ $S_n = \frac{8}{9} \left[ \frac{10(10^n - 1)}{9} - n \right] = \frac{80}{81}(10^n - 1) - \frac{8n}{9}$.