# How to find the number of continued fraction from a periodic representation?

Problem
Find the number that represented by $[2,2,2 \ldots]$

I know it wasn't difficult, but I was absent the last two classes. So I just want to make sure that I got it right.

My attempt was,
Consider $x = [2;\overline{2}] \implies [2;x]$ Hence, $x = 2 + \dfrac{1}{x} \Leftrightarrow x = \dfrac{2x + 1}{x} \Leftrightarrow x^2 - 2x - 1 = 0$. Solving the quadratic equation we have $x = \sqrt{2} + 1$ since $x > 0$. Therefore $[2,2,2, \ldots] = \sqrt{2} + 1$

Is it correct?

Thank you,

-

2
That looks right to me. – yunone May 2 '11 at 3:45

2
Yes that looks right, and is in fact a different proof that $\sqrt{2}$ is irrational. – Aryabhata May 2 '11 at 3:51

2
Yes - now try something a little harder, like $[2,3,4,3,4,3,4,3,4,\dots]$, with the $3,4$ repeating. – Gerry Myerson May 2 '11 at 4:12
@all: Thanks a lot. – Chan May 2 '11 at 4:24
If you put a number into Wolfram Alpha it'll give you back its continued fraction representation. This isn't the direction you're trying to go, but it'll help you check your work. – Michael Lugo May 2 '11 at 16:05
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 9, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9688932299613953, "perplexity_flag": "middle"}