Express the recurring decimal 0.1bar576 as rational number using the concept of geometric series?

1 Answer
Aug 26, 2017

0.1bar(576)=35/222

Explanation:

0.1bar(576)

= 0.1576576576576576....

= 1/10+576/10000+576/10000000+576/10000000000+576/10000000000000+........

= 1/10+576/10^4+576/10^7+576/10^10+576/10^13+......

= 1/10+576/10^4(1+10^(-3)+10^(-6)+10^(-9)+.......)

= 1/10+576/10000(1/(1-10^(-3)))

= 1/10+576/10000xx1000/999

= 1/10+576/9990

= (999+576)/9990

= 1575/9990

= 105/666

= 35/222