Question

Q 10.4

Answer 1

Required sum = 0.5 + 0.55 + 0.555 +..........upto n terms
= 5(0.1 + 0.11 + 0.111 +..........upto n terms

=${\displaystyle \frac{5}{9}}$(0.9 + 0.99 + 0.999 +..........upto n terms

=${\displaystyle \frac{5}{9}}$[(1 - 0.1) + (1 - 0.01) +(1 -  0.001) +..........upto n terms]

=${\displaystyle \frac{5}{9}}$[(1 + 1 + 1 +..........upto n terms) - ( 0.1 + 0.01 + 0.001 +..........upto n terms)]

=${\displaystyle \frac{5}{9}[(1+1+1+..........\text{upto n terms})-(\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+..........\text{upto n terms})]}$

${\displaystyle =\frac{5}{9}[n-\frac{\frac{1}{10}(1-\frac{1}{{10}^n})}{1-\frac{1}{10}}]}$   ${\displaystyle [\because r=\frac{1}{10}<1]}$

${\displaystyle =\frac{5}{9}[n-\frac{10}{9}\times \frac{1}{10}(1-\frac{1}{{10}^n})]}$

${\displaystyle =\frac{5}{9}[n-\frac{1}{9}(1-\frac{1}{{10}^n})]}$