Why do divisions like 1/98 and 1/998 give us numbers continuously being multiplied by two each time in decimal form?

Why do divisions like 1/98 and 1/998 give us numbers continuously being multiplied by two each time in decimal form?

Nice observation. We can write

\frac{1}{98} = \frac{1}{100 - 2} = \frac{1}{100} \frac{1}{1 - \frac{2}{100}}

Now for $| x | < 1$ , $\frac{1}{1 - x} = 1 + x + x^{2} + x^{3} + \dots$

Apply that result to the expression above and we have

\frac{1}{98} = \frac{1}{100} (1 + (\frac{2}{100}) + {(\frac{2}{100})}^{2} + {(\frac{2}{100})}^{3} + \dots)

That gives the pattern you observe.

You can make a similar analysis for $\frac{1}{998}$ .

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