Question

Please help me on my hw I need help on #2

Answer 1

Given:

The number is,

`5.232323\ldots\text{.}`

To express the given number into fraction . it means in the form,

`(a)/(b)`

We can express the given number into geometric series as,

`\begin{gathered} 5.232323\ldots=5+(23)/(100)+(23)/(10000)+(23)/(100000)+\text{.}\ldots\ldots \\ =5+(23)/(100)+23((1)/(100))^2+23((1)/(100))^3+\text{.}\ldots\ldots\text{.}\mathrm{}(1) \\ (23)/(100)+23((1)/(100))^2+23((1)/(100))^3+\text{.}\ldots=-23+\sum ^(\infty)_(n\mathop=1)23((1)/(100))^(n-1) \\ =-23+(23)/(1-(1)/(100)) \\ =-23+(23(100))/(99) \\ =-23+(2300)/(99) \\ =(-2277+2300)/(99) \\ =(23)/(99) \end{gathered}`

Now, equation (1) becomes,

`5+(23)/(99)=(518)/(99)`

Answer:

`(518)/(99)`