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Limit of consecutive quotients

Johannes Kepler observed that the ratio of consecutive Fibonacci numbers converges. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost", and concluded that these ratios approach the golden ratio φ: {\displaystyle \varphi \colon } lim n → ∞ F n + 1 F n = φ. {\displaystyle \lim _{n\to \infty }{\frac {F_{n+1}}{F_{n}}}=\varphi.}

This convergence holds regardless of the starting values U 0 {\displaystyle U_{0}} and U 1 {\displaystyle U_{1}}, unless U 1 = − U 0 / φ {\displaystyle U_{1}=-U_{0}/\varphi }. This can be verified using Binet's formula. For example, the initial values 3 and 2 generate the sequence 3, 2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555,.... The ratio of consecutive terms in this sequence shows the same convergence towards the golden ratio.

In general, lim n → ∞ F n + m F n = φ m {\displaystyle \lim _{n\to \infty }{\frac {F_{n+m}}{F_{n}}}=\varphi ^{m}}, because the ratios between consecutive Fibonacci numbers approaches φ {\displaystyle \varphi }.