Question

The sum of the first 10 terms of the sequence $8,88,888,8888, \ldots$ 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)

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