Question

Sum upto $n$ the series
$.7+.77+.777+......$

Answer 1

$S=.7+.77+.777+..n$ terms
$=\frac{7}{9}[.9+.99+.999+....]$
$=\frac{7}{9}[(1-\frac{1}{10})+(1-\frac{1}{100})+(1-\frac{1}{1000})+........]$
$=\frac{7}{9}[(1+1+1+...)-(\frac{1}{10}+\frac{1}{{10}^2}\frac{1}{{10}^3}+..)]$
$=\frac{7}{9}[n-\frac{1}{10}.\frac{1-\left(1/{10}^n\right)}{1-\left(1/10\right)}]=\frac{7n}{9}-\frac{7}{81}(1-\frac{1}{{10}^n})$