# How does the golden ratio relate to the Fibonacci sequence?

Nov 7, 2015

The ratio between successive terms of the Fibonacci sequence tends towards $\phi$, or the golden ratio.

#### Explanation:

If you take two consecutive terms in the Fibonacci Sequence, or indeed any given sequence, and divide one of the terms by the one that comes before it, then you get a number that will get closer and closer to $\phi$, or $1.618033988 \ldots$

An example is the division $34 \div i \mathrm{de} 21$, which gives the answer $1.61904762$. Going even further to $17711 \div i \mathrm{de} 10946$, which gives $1.61803399$.

In addition, there is a direct formula for ${F}_{n}$ in terms of $\phi$:

${F}_{n} = \frac{{\phi}^{n} - {\left(- \phi\right)}^{-} n}{\sqrt{5}}$
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 10, "mathjax_inline_tex": 1, "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, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://socratic.org/questions/how-does-the-golden-ratio-relate-to-the-fibonacci-sequence#185342", "fetch_time": 1680106287000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2023-14/segments/1679296949009.11/warc/CC-MAIN-20230329151629-20230329181629-00158.warc.gz", "warc_record_offset": 566985583, "warc_record_length": 6147, "token_count": 213, "char_count": 760, "score": 4.09375, "int_score": 4, "crawl": "CC-MAIN-2023-14", "snapshot_type": "latest", "language": "en", "language_score": 0.8595499396324158, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00060-of-00064.parquet"}