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
Shwetank Mauria
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 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#

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