Binet's formula

Theorem

The following formula holds: $$F_n = \dfrac{\phi^n - (-\phi)^{-n}}{\sqrt{5}},$$ where $F_n$ denotes a Fibonacci number and $\phi$ denotes the golden ratio.

References

• {{ #if: |{{{2}}}|John H. Halton}}{{#if: |{{#if: |, [[Mathematician:{{{author2}}}|{{ #if: |{{{2}}}|{{{author2}}}}}]]{{#if: |, [[Mathematician:{{{author3}}}|{{ #if: |{{{2}}}|{{{author3}}}}}]]{{#if: |, [[Mathematician:{{{author4}}}|{{ #if: |{{{2}}}|{{{author4}}}}}]]{{#if: |, [[Mathematician:{{{author5}}}|{{ #if: |{{{2}}}|{{{author5}}}}}]] and [[Mathematician:{{{author6}}}|{{ #if: |{{{2}}}|{{{author6}}}}}]]| and [[Mathematician:{{{author5}}}|{{ #if: |{{{2}}}|{{{author5}}}}}]]}}| and [[Mathematician:{{{author4}}}|{{ #if: |{{{2}}}|{{{author4}}}}}]]}}| and [[Mathematician:{{{author3}}}|{{ #if: |{{{2}}}|{{{author3}}}}}]]}}| and [[Mathematician:{{{author2}}}|{{ #if: |{{{2}}}|{{{author2}}}}}]]}}|}}: [[Paper:John H. Halton/On a General Fibonacci Identity{{#if: |/Volume {{{volume}}}|}}{{#if: |/{{{edpage}}}}}|On a General Fibonacci Identity{{#if: |: Volume {{{volume}}}|}}{{#if: |: {{{eddisplay}}} (1965)| ({{#if: |{{{ed}}} ed., }}1965)}}]]{{#if: |, {{{publisher}}}|}}{{#if: |, ISBN {{{isbn}}}|}}{{#if: Fibonacci numbers | ... (previous)|}}{{#if: F(-n)=(-1)^(n+1)F(n) | ... (next)|}}{{#if: |: Entry: {{#if: |[[{{{entryref}}}|{{{entry}}}]]|{{{entry}}}}}|}}
• {{ #if: |{{{2}}}|Maruti Ram Murty}}{{#if: |{{#if: |, [[Mathematician:{{{author2}}}|{{ #if: |{{{2}}}|{{{author2}}}}}]]{{#if: |, [[Mathematician:{{{author3}}}|{{ #if: |{{{2}}}|{{{author3}}}}}]]{{#if: |, [[Mathematician:{{{author4}}}|{{ #if: |{{{2}}}|{{{author4}}}}}]]{{#if: |, [[Mathematician:{{{author5}}}|{{ #if: |{{{2}}}|{{{author5}}}}}]] and [[Mathematician:{{{author6}}}|{{ #if: |{{{2}}}|{{{author6}}}}}]]| and [[Mathematician:{{{author5}}}|{{ #if: |{{{2}}}|{{{author5}}}}}]]}}| and [[Mathematician:{{{author4}}}|{{ #if: |{{{2}}}|{{{author4}}}}}]]}}| and [[Mathematician:{{{author3}}}|{{ #if: |{{{2}}}|{{{author3}}}}}]]}}| and [[Mathematician:{{{author2}}}|{{ #if: |{{{2}}}|{{{author2}}}}}]]}}|}}: [[Paper:Maruti Ram Murty/The Fibonacci Zeta Function{{#if: |/Volume {{{volume}}}|}}{{#if: |/{{{edpage}}}}}|The Fibonacci Zeta Function{{#if: |: Volume {{{volume}}}|}}{{#if: |: {{{eddisplay}}} (1976)| ({{#if: |{{{ed}}} ed., }}1976)}}]]{{#if: |, {{{publisher}}}|}}{{#if: |, ISBN {{{isbn}}}|}}{{#if: Fibonacci zeta function | ... (previous)|}}{{#if: Fibonacci zeta in terms of a sum of binomial coefficients | ... (next)|}}{{#if: |: Entry: {{#if: |[[{{{entryref}}}|{{{entry}}}]]|{{{entry}}}}}|}}