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Find the sum to $n$ terms of the sequence, $8,88,888,8888 \ldots$
$\frac{80}{81}\left(10^{n}-1\right)-\frac{8}{9} n$
$\frac{80}{81}\left(10^{n}-1\right)-\frac{8}{9} n$
$\frac{80}{81}\left(10^{n}-1\right)-\frac{8}{9} n$
$\frac{80}{81}\left(10^{n}-1\right)-\frac{8}{9} n$
Solution
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
$S_{n}=8+88+888+8888+\ldots \ldots$ to $n$ terms
$=\frac{8}{9}[9+99+999+9999+\ldots \ldots . . $ to $ n $ terms $]$
$=\frac{8}{9}[(10+10^{2}+\ldots \ldots . n \text { terms })$
$-(1+1+1+\ldots . . n \text { terms })]$
$=\frac{8}{9}\left[\frac{10\left(10^{n}-1\right)}{10-1}-n\right]$
$=\frac{8}{9}\left[\frac{10\left(10^{n}-1\right)}{9}-n\right]$
$=\frac{80}{81}\left(10^{n}-1\right)-\frac{8}{9} n$
