Question Number 135797 by benjo_mathlover last updated on 16/Mar/21
$$ \\ $$ How do you solve f(3) when f(-1) = -1 and f(0) =1 and f(x) =f(x-1)-2f(x-2)?\\n
Answered by liberty last updated on 17/Mar/21
$${f}\left({x}\right)={f}\left({x}−\mathrm{1}\right)−\mathrm{2}{f}\left({x}−\mathrm{2}\right) \\ $$ $${replace}\:{x}−\mathrm{2}\:{by}\:{x}\:{we}\:{get} \\ $$ $$\mathrm{2}{f}\left({x}\right)={f}\left({x}+\mathrm{1}\right)−{f}\left({x}+\mathrm{2}\right) \\ $$ $$ \\ $$ $$\left(\mathrm{1}\right)\:{x}=−\mathrm{1}\Rightarrow\mathrm{2}{f}\left(−\mathrm{1}\right)={f}\left(\mathrm{0}\right)−{f}\left(\mathrm{1}\right) \\ $$ $$\Rightarrow−\mathrm{2}=\mathrm{1}−{f}\left(\mathrm{1}\right);\:{\color{mathred}{f}}\color{mathred}{\left(}\mathrm{\color{mathred}{1}}\color{mathred}{\right)}\color{mathred}{=}\color{mathred}{\:}\mathrm{\color{mathred}{3}} \\ $$ $$ \\ $$ $$\left(\mathrm{2}\right){x}=\mathrm{0}\Rightarrow\mathrm{2}{f}\left(\mathrm{0}\right)={f}\left(\mathrm{1}\right)−{f}\left(\mathrm{2}\right) \\ $$ $$\Rightarrow\mathrm{2}\:=\:\mathrm{3}−{f}\left(\mathrm{2}\right);\:{\color{mathred}{f}}\color{mathred}{\left(}\mathrm{\color{mathred}{2}}\color{mathred}{\right)}\color{mathred}{=}\mathrm{\color{mathred}{1}} \\ $$ $$ \\ $$ $$\left(\mathrm{3}\right){x}=\mathrm{1}\Rightarrow\mathrm{2}{f}\left(\mathrm{1}\right)={f}\left(\mathrm{2}\right)−{f}\left(\mathrm{3}\right) \\ $$ $$\Rightarrow\mathrm{6}\:=\:\mathrm{1}−{f}\left(\mathrm{3}\right);\:{\color{mathred}{f}}\color{mathred}{\left(}\mathrm{\color{mathred}{3}}\color{mathred}{\right)}\color{mathred}{=}\color{mathred}{−}\mathrm{\color{mathred}{5}} \\ $$