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[parent] values of $\frac{F_n}{F_{n - 1}}$ for $1 < n < 26$ (Example)

The golden ratio $ \phi$ is an irrational number, approximately 1.6180339887498948482045868. Dividing one Fibonacci number by the previous gives better and better approximations to the golden ratio the bigger the numbers get. One division gets a little under, then the next a little over, as the table below shows. 144 divided by 89 is probably good enough for most practical purposes.

$ F_n$ $ F_{n - 1}$ $ \frac{F_n}{F_{n - 1}}$ to 20 decimal places Approximation error
1 1 1.00000000000000000000 -0.61803398874989484820
2 1 2.00000000000000000000 0.38196601125010515179
3 2 1.50000000000000000000 -0.11803398874989484820
5 3 1.66666666666666666666 0.04863267791677181846
8 5 1.60000000000000000000 -0.01803398874989484820
13 8 1.62500000000000000000 0.00696601125010515179
21 13 1.61538461538461538461 -0.00264937336527946358
34 21 1.61904761904761904761 0.00101363029772419941
55 34 1.61764705882352941176 -0.00038692992636543643
89 55 1.61818181818181818181 0.00014782943192333361
144 89 1.61797752808988764044 -0.00005646066000720775
233 144 1.61805555555555555555 0.00002156680566070735
377 233 1.61802575107296137339 -0.00000823767693347481
610 377 1.61803713527851458885 0.00000314652861974065
987 610 1.61803278688524590163 -1.20186464894656524257
1597 987 1.61803444782168186423 0.00000045907178701603
2584 1597 1.61803381340012523481 -0.00000017534976961338
4181 2584 1.61803405572755417956 0.00000066977659331361
6765 4181 1.61803396316670652953 -0.00000002558318831866
10946 6765 1.61803399852180339985 0.00000000977190855164
17711 10946 1.61803398501735793897 -0.00000000373253690923
28657 17711 1.61803399017559708655 0.00000000142570223835
46368 28657 1.61803398820532505147 -0.00000000054456979673
75025 46368 1.61803398895790200138 0.00000000020800715317



"values of $\frac{F_n}{F_{n - 1}}$ for $1 < n < 26$" is owned by PrimeFan.
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Cross-references: decimal places, division, approximations, Fibonacci number, irrational number, golden ratio

This is version 1 of values of $\frac{F_n}{F_{n - 1}}$ for $1 < n < 26$, born on 2007-07-14.
Object id is 9767, canonical name is ValuesOfFracF_nF_n1For1N26.
Accessed 342 times total.

Classification:
AMS MSC: 11B39 (Number theory :: Sequences and sets :: Fibonacci and Lucas numbers and polynomials and generalizations)
 40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences)
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