Question

Write the following decimal as a fraction by writing the decimal as a geometric series?

Write the following decimal as a fraction by writing the decimal as a geometric series.
4.13131313...

If it was, for example, .13131313... it would be 13/99 but this problem has a number before it. What are the steps to solve this problem instead?

Answer 1

`4.1313131313......=4 13/99`

Explanation:

We can write `4.1313131313......` as

`4+1/10^1+3/10^2+1/10^3+3/10^4+1/10^5+3/10^6+1/10^7+3/10^7....`

or `4+13/10^2+13/10^4+13/10^6+13/10^8+.........`

Leaving aside `4`, theremaiing part is a geometric infinite series, with first term as `13/100` and common ratio as`1/100` and as the series is infinite(whose sum is `a/(1-r)` where `a` is first term and `r` is common ratio) and we have

`4.1313131313......`

= `4+(13/100)/(1-1/100)`

= `4+(13/100)/(99/100)`

= `4+13/9`

= `4 13/99`