Question Number 194693 by MM42 last updated on 13/Jul/23
$${if}\:\:\:{f}_{{n}} ={f}_{{n}−\mathrm{1}} +{f}_{{n}−\mathrm{2}} \:\:;\:\:{f}_{\mathrm{1}} ={f}_{\mathrm{2}} =\mathrm{1} \\ $$$${then}\:\:\:{prove}\:{that}\:\:\:\mathrm{5}\mid{f}_{\mathrm{5}{n}} \:\: \\ $$
Answered by Frix last updated on 13/Jul/23
$${f}_{\mathrm{1}} ={k}_{\mathrm{1}} \\ $$$${f}_{\mathrm{2}} ={k}_{\mathrm{2}} \\ $$$${f}_{{n}} ={f}_{{n}−\mathrm{1}} +{f}_{{n}−\mathrm{2}} \forall{n}\geqslant\mathrm{3} \\ $$$$\Rightarrow \\ $$$${f}_{{n}} =\mathrm{3}{f}_{{n}−\mathrm{5}} +\mathrm{5}{f}_{{n}−\mathrm{4}} \forall{n}\geqslant\mathrm{6} \\ $$$$\mathrm{5}\mid{f}_{{n}−\mathrm{5}} \:\Leftrightarrow\:\mathrm{5}\mid{f}_{{n}} \forall{n}\geqslant\mathrm{6} \\ $$
Commented by MM42 last updated on 13/Jul/23
$${that}'{s}\:{right} \\ $$