Sequences and Series: Sum of Geometric Progression with Repeating Digits

$$S_n = 7 + 77 + 777 + \dots + (\text{to } n \text{ terms})$$ $$S_n = 7(1 + 11 + 111 + \dots + \text{to } n \text{ terms})$$ $$S_n = \frac{7}{9}(9 + 99 + 999 + \dots + \text{to } n \text{ terms})$$

$$S_n = \frac{7}{9} \left[ (10 - 1) + (10^2 - 1) + (10^3 - 1) + \dots + (10^n - 1) \right]$$ $$S_n = \frac{7}{9} \left[ (10 + 10^2 + 10^3 + \dots + 10^n) - (1 + 1 + 1 + \dots \text{ to } n \text{ times}) \right]$$

For GP $10 + 10^2 + \dots + 10^n$: $a = 10, r = 10$. $$\text{Sum} = \frac{a(r^n - 1)}{r - 1} = \frac{10(10^n - 1)}{10 - 1} = \frac{10(10^n - 1)}{9}$$ $$\text{Subtracting } n: \quad S_n = \frac{7}{9} \left[ \frac{10(10^n - 1)}{9} - n \right]$$

$$S_n = \frac{70}{81}(10^n - 1) - \frac{7n}{9}$$