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Find the sum of the following series up to n terms:
$5+55+555+\ldots$
$\frac{50}{81}\left(10^{n}-1\right)-\frac{5 n}{9}$
$\frac{50}{81}\left(10^{n}-1\right)-\frac{5 n}{9}$
$\frac{50}{81}\left(10^{n}-1\right)-\frac{5 n}{9}$
$\frac{50}{81}\left(10^{n}-1\right)-\frac{5 n}{9}$
Solution
$5+55+555+\ldots$
Let $S_{n}=5+55+555+\ldots .$ to $n$ terms
$=\frac{5}{9}[9+99+999+\ldots \ldots \text { to } n \text { terms }]$
$=\frac{5}{9}\left[(10-1)+\left(10^{2}-1\right)+\left(10^{3}-1\right)+\ldots \text { to } n \text { terms }\right]$
$=\frac{5}{9}[\left(10+10^{2}+10^{3}+\text { to } n \text { terms }\right)$
$-(1+1+\ldots \text { to } n \text { terms })]$
$=\frac{5}{9}\left[\frac{10\left(10^{n}-1\right)}{10-1}-n\right]$
$=\frac{5}{9}\left[\frac{10\left(10^{n}-1\right)}{9}-n\right]$
$=\frac{50}{81}\left(10^{n}-1\right)-\frac{5 n}{9}$
