Writing $2.025252525252$ (repeating) as a fraction with Calculus(Series)

Question

My math teacher assigned this question and I am unsure on how to solve it with series from Calc BC(no calculator also).

Answer 1

Hint:

Evaluate $$2.0 + 0.02525 \ldots = 2+0.1(0.2525\ldots)=2+0.1(25)\sum_{i=1}^\infty \left(\frac{1}{100}\right)^i$$

Answer 2

Let $x=2.0\overline{25}$.

Then $1000x-10x=2025.\overline{25}-20.\overline{25}=2005$

so $990x=2005$.

Can you take it from here?

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Alternatively, using geometric series,

$x=2+\dfrac{25}{1000}+\dfrac{25}{10^5}+\dfrac{25}{10^7}+\cdots$

$=2+\dfrac{25/1000}{1-1/100}=2+\dfrac{25}{990}$

Answer 3

The trick with repeating fractions is to multiply by some power of $10$ then subtract off the repeating part. I solve them as follows.

If $x=2.0252525...$ then $100x=202.5252525...$ and so $100x-x=99x=200.5$ and it's not hard to finish from there.