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!
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
http://www.flickr.com/photos/dwread/6976…
http://www.flickr.com/photos/dwread/6976…