Question

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

Answer 1

`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`