Recurrence Relations

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1. solution Find the limiting ratio \lim _{{n\rightarrow \infty }}{\frac  {x_{{n+1}}}{x_{n}}}\, for the recurrence relation x_{n}=x_{{n-1}}+x_{{n-2}}\,.

2. solution Let x_{n}:=\left\{{\begin{matrix}x_{{n-1}}+x_{{n-2}}&{\mbox{if }}n>1\\x_{1}\in {\mathbb  {N}}&{\mbox{if }}n=1\\x_{0}\in {\mathbb  {N}}&{\mbox{if }}n=0\\0&{\mbox{if }}n<0\end{matrix}}\right.     Show that x_{n}=x_{{1}}F_{{n-1}}+x_{{0}}F_{{n-2}}\,\! where F_{n}\,\! is the n-th Fibonacci number   (F_{0}=F_{1}=1)

3. solution Let x_{n}:=\left\{{\begin{matrix}1+{\frac  {1}{x_{{n-1}}}}&{\mbox{if }}n>0\\x_{0}\in {\mathbb  {N}}&{\mbox{if }}n=0\\0&{\mbox{if }}n<0\end{matrix}}\right.     Show that x_{n}={\frac  {x_{{0}}F_{{n}}+F_{{n-1}}}{x_{{0}}F_{{n-1}}+F_{{n-2}}}}\,\! where F_{n}\,\! is the n-th Fibonacci number   (F_{0}=F_{1}=1{\mbox{ and }}F_{{(n<0)}}=0)

4. solution Let x_{n}:=\left\{{\begin{matrix}x_{{n-1}}*x_{{n-2}}&{\mbox{if }}n>1\\x_{1}\in {\mathbb  {N}}&{\mbox{if }}n=1\\x_{0}\in {\mathbb  {N}}&{\mbox{if }}n=0\\0&{\mbox{if }}n<0\end{matrix}}\right.     Show that x_{n}=x_{{1}}^{{F_{{n}}}}*x_{{2}}^{{F_{{n-1}}}}\,\! where F_{n}\,\! is the n-th Fibonacci number   (F_{0}=F_{1}=1{\mbox{ and }}F_{{(n<0)}}=0)

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