Question

The sum of the first 10 terms of the sequence $8,88,888,8888,\dots$ is ____

• A. $\frac{8}{9}\left[\frac{1}{9}\left(10^{10}-1\right)\right]$
• B. $\frac{8}{9}\left[\frac{10}{9}\left(10^{10}-10\right)\right]$
• C. $\frac{8}{9}\left[\left(10^{10}-1\right)-10\right]$
• D. $\frac{8}{9}\left[\frac{10}{9}\left(10^{10}-1\right)-9\right]$

Answer 1

Answer: (B)

$S_{10}=8+88+888+8888+\dots 10\text{ terms }$
$S_{10}=\frac{8}{9}[9+99+999+9999+\dots 10\text{ terms }]$
$=\frac{8}{9}[(10-1)+(100-1)+(1000-1)+\dots +\dots 10\text{ terms }]$
$=\frac{8}{9}[10^1+10^2+10^3+\dots (10\text{ terms })-$$(1+1+1+\dots 10$ terms $)]$
$=\frac{8}{9}\left[\frac{10\left(10^{10}-1\right)}{10-1}-10\right]$
$=\frac{8}{9}[\frac{10}{9}\left(10^{10}-10\right]$