Question

Express the repeating decimal as a fraction in lowest terms.
0.22=22/100+22/10,000+22/1,000,000+....

Answer 1

So you want to write \(0.\overline{2}\) as a fraction, right?

Answer 2

One way to do it is by writing it as a sum of fractions like you did:
\[0.\overline{2}=\frac{2}{10}+\frac{2}{100}+\frac{2}{1000}+\cdots\]
which is an infinite geometric series,
\[\large 0.\overline{2}=\sum_{n=1}^\infty2\left(\frac{1}{10}\right)^n\]
Have you worked with infinite series yet?

Answer 3

Yes, i want to write it as a fraction in lowest terms

Answer 4

Okay, but before I continue, do you know what infinite series are? There's another somewhat simpler method we can try if that's the case.

Answer 5

is it infinity numbers?

Answer 6

"Series" refers to a summation. "Infinite series" means we're adding up an infinite number of numbers.
\[0.\overline{2}=\frac{2}{10}+\frac{2}{100}+\frac{2}{1000}+\frac{2}{10000}+\frac{2}{100000}+\cdots\]
The \(\cdots\) means you go on and on with the pattern. The \(\sum\) form is a concise way of expressing this infinite sum. If you've never used the symbol \(\sum\) before, it sounds like you haven't learned about infinite series yet.

Answer 7

yes, i have used the symbol before (sum)

Answer 8

Okay, so do you know that
\[\large\sum_{n=0}^\infty ar^n\]
is a geometric series with common ratio \(r\)? And that
\[\large\sum_{n=0}^\infty ar^n=\frac{a}{1-r}~~\text{for}~~|r|<1~~?\]

Answer 9

We just started that chapter yesterday, didn't really understand it yet.

Answer 10

Which part, specifically, do you not understand? I'll try to explain anything you might not get.

Answer 11

how to write the equation into lowest terms

Answer 12

I had not seen that formula before

Answer 13

\[\sum_{i=n}^{4}1/3i\]

Answer 14

is that the same, because that's all we did so far? like find the indicated sum

Answer 15

That's a finite sum. It's kind of the same but there's not infinite number of terms.

Answer 16

Sorry about the delayed responses. I'm a bit busy. If you have the time, feel free to watch this video. It'll explain everything I wanted to touch on: https://www.khanacademy.org/math/precalculus/seq_induction/infinite-geometric-series/v/repeating-decimal-geometric-series