I'm working on converting numbers in various bases and one question asks to convert $.2525...$ from decimal to octal. I know that the answer is $1/3$ and that it is necessary to use the infinite series representation of this decimal to find the answer, but I seem to be running into trouble with my conversions, or there is something that I'm missing.
2026-04-19 16:20:43.1776615643
Base conversion using geometric series
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As JiK noticed, the intended direction is: from octal to decimal. In the octal system, $$0.25_8 = \frac{2}{8}+\frac{5}{64} = \frac{21}{64}$$ The infinite octal expansion $0.2525252525\dots_8$ is
$$0.25_8+ \frac{1}{64}0.25_8+\frac{1}{64^2}0.25_8 +\dots = \frac{21}{64} \frac{1}{1-1/64} = \frac{21}{63}$$ which is indeed $1/3$.