Problem Statement
Find the sum to $n$ terms of the series:
$$S_n = 8 + 88 + 888 + 8888 + \dots \text{ to } n \text{ terms}$$
Verified Solution & Marking Scheme
Factor out Common Term 8
$$S_n = 8(1 + 11 + 111 + 1111 + \dots \text{ to } n \text{ terms})$$
Multiply and Divide by 9
$$S_n = \frac{8}{9} (9 + 99 + 999 + 9999 + \dots \text{ to } n \text{ terms})$$
Express Terms as Powers of 10 Minus 1
$$9 = 10 - 1, \quad 99 = 10^2 - 1, \quad 999 = 10^3 - 1, \dots$$
$$S_n = \frac{8}{9} [(10 - 1) + (10^2 - 1) + (10^3 - 1) + \dots + (10^n - 1)]$$
$$= \frac{8}{9} [(10 + 10^2 + 10^3 + \dots + 10^n) - (1 + 1 + 1 + \dots + 1)]$$
$$= \frac{8}{9} [(10 + 10^2 + \dots + 10^n) - n]$$
Apply Geometric Progression Sum Formula
The series $10 + 10^2 + \dots + 10^n$ is a G.P. with first term $a = 10$, common ratio $r = 10$, and $n$ terms:
$$\sum_{k=1}^n 10^k = \frac{a(r^n - 1)}{r - 1} = \frac{10(10^n - 1)}{10 - 1} = \frac{10(10^n - 1)}{9}$$
$$S_n = \frac{8}{9} \left[ \frac{10(10^n - 1)}{9} - n \right] = \frac{80}{81}(10^n - 1) - \frac{8n}{9}$$.