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

=`5/9`(0.9 + 0.99 + 0.999 +..........upto n terms

=`5/9`[(1 - 0.1) + (1 - 0.01) +(1 -  0.001) +..........upto n terms]

=`5/9`[(1 + 1 + 1 +..........upto n terms) - ( 0.1 + 0.01 + 0.001 +..........upto n terms)]

=`5/9[(1 + 1 + 1 +.........."upto n terms") - (1/10 + 1/100 + 1/1000 +.........."upto n terms")]`

`=5/9[n-(1/10(1-1/10^n))/(1-1/10)]`   `[becauser = 1/10<1]`

`=5/9[n-10/9xx1/10(1-1/10^n)]`

`=5/9[n-1/9(1-1/10^n)]`