Given x=.31313131... in base 5, find the value of x (in base10) expressed as a fraction in lowest terms
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Given x=.31313131... in base 5, find the value of x (in base10) expressed as a fraction in lowest terms

[From: ] [author: ] [Date: 12-04-29] [Hit: ]
-Note that, in base 10,x = 3/5^1 + 1/5^2 + 3/5^3 + 1/5^4 + ...= (3/5^1 + 3/5^3 + 3/5^5 + .......
Please explain your steps. I was trying to convert the base 5 decimal into a base 10 decimal, and then to a fraction, but I'm stuck with the conversion. Thanks a bunch!

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Note that, in base 10, we have:
x = 3/5^1 + 1/5^2 + 3/5^3 + 1/5^4 + ...
= (3/5^1 + 3/5^3 + 3/5^5 + ...) + (1/5^2 + 1/5^4 + 1/5^6 + ...).

The above follows from the fact that, in base-5, numbers are represented in powers of 5 instead of powers of 10.

Both of the above series are infinite geometric series. The first series has a first term of 3/5 and a common ratio of 1/5^2 = 1/25, so:
3/5^1 + 3/5^3 + 3/5^5 + ... = (3/5)/(1 - 1/25) = 15/(25 - 1) = 15/24.

In a similar fashion, the second series has a first term of 1/5^2 = 1/25 and a common ratio of 1/25, so:
1/5^2 + 1/5^4 + 1/5^6 + ... = (1/25)/(1 - 1/25) = 1/(25 - 1) = 1/24.

Therefore, x = 15/24 + 1/24 = 16/24 = 2/3 in base-10.

I hope this helps!

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I get ⅔
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