Find the sum to $n$ terms of the sequence, $8,88,888,8888 \ldots$
The given sequence is $8,88,888,8888 \ldots$
This sequence is not a G.P. However, it can be changed to G.P. by writing the terms as
$S_{n}=8+88+888+8888+\ldots \ldots \ldots \ldots \ldots \ldots$ to $n$ terms
$=\frac{8}{9}[9+99+999+9999+\ldots \ldots \ldots .$ to $n$ terms $]$
$=\frac{8}{9}\left[(10-1)+\left(10^{2}-1\right)+\left(10^{3}-1\right)+\left(10^{4}-1\right)+\ldots \ldots . .\right.$ to $n$ terms $]$
$=\frac{8}{9}\left[\left(10+10^{2}+\ldots . . n\right.\right.$ terms $)-(1+1+1+\ldots . n$ terms $\left.)\right]$
$=\frac{8}{9}\left[\frac{10\left(10^{n}-1\right)}{10-1}-n\right]$
$=\frac{8}{9}\left[\frac{10\left(10^{n}-1\right)}{9}-n\right]$
$=\frac{80}{81}\left(10^{n}-1\right)-\frac{8}{9} n$
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