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?

1 Answer
Dec 31, 2017

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 as1/100 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