# values of $\frac{F_{n}}{F_{n-1}}$ for $1

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 1.00000000000000000000 -0.61803398874989484820 2.00000000000000000000 0.38196601125010515179 1.50000000000000000000 -0.11803398874989484820 1.66666666666666666666 0.04863267791677181846 1.60000000000000000000 -0.01803398874989484820 1.62500000000000000000 0.00696601125010515179 1.61538461538461538461 -0.00264937336527946358 1.61904761904761904761 0.00101363029772419941 1.61764705882352941176 -0.00038692992636543643 1.61818181818181818181 0.00014782943192333361 1.61797752808988764044 -0.00005646066000720775 1.61805555555555555555 0.00002156680566070735 1.61802575107296137339 -0.00000823767693347481 1.61803713527851458885 0.00000314652861974065 1.61803278688524590163 -1.20186464894656524257 1.61803444782168186423 0.00000045907178701603 1.61803381340012523481 -0.00000017534976961338 1.61803405572755417956 0.00000066977659331361 1.61803396316670652953 -0.00000002558318831866 1.61803399852180339985 0.00000000977190855164 1.61803398501735793897 -0.00000000373253690923 1.61803399017559708655 0.00000000142570223835 1.61803398820532505147 -0.00000000054456979673 1.61803398895790200138 0.00000000020800715317
Title values of $\frac{F_{n}}{F_{n-1}}$ for $1 ValuesOffracFnFn1For1N26 2013-03-22 17:23:54 2013-03-22 17:23:54 PrimeFan (13766) PrimeFan (13766) 4 PrimeFan (13766) Example msc 40A05 msc 11B39