Question

Find the nth term of the sequence $0.7,0.77,\mathrm{0.777....}$

Answer 1

$0.7+0.77+\mathrm{0.777....}$

$=\frac{7}{9}\{0.9+0.99+0.999+......\}$

$=\frac{7}{9}\left[\right(1-0.1)+(1-0.01)+(1-0.001)+........]$

$=\frac{7}{9}\left[\right(1+1+......+n)-(\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+..........+n\left)\right]$

$=\frac{7}{9}\left[\frac{n-0.1\times (1-(0.1{)}^n)}{1-0.1}\right]$

$=\frac{7}{81}[9n-1+{10}^{-n}]$