Question Number 81910 by mr W last updated on 16/Feb/20
$${a}_{\mathrm{1}} =\mathrm{1} \\ $$$${a}_{\mathrm{2}} =\mathrm{2} \\ $$$${a}_{{n}+\mathrm{1}} =\left({n}+\mathrm{1}\right){a}_{{n}} −\mathrm{2}{a}_{{n}−\mathrm{1}} \\ $$$${find}\:{a}_{{n}} =? \\ $$
Commented by jagoll last updated on 16/Feb/20
$${a}_{\mathrm{3}} \:=\:\mathrm{3}{a}_{\mathrm{2}} −\mathrm{2}{a}_{\mathrm{1}} \:=\:\mathrm{6}−\mathrm{2}=\mathrm{4} \\ $$$${a}_{\mathrm{4}\:} =\:\mathrm{4}{a}_{\mathrm{3}} −\mathrm{2}{a}_{\mathrm{2}} \:=\:\mathrm{16}−\mathrm{4}=\mathrm{12} \\ $$$${a}_{\mathrm{5}} \:=\:\mathrm{5}{a}_{\mathrm{4}} −\mathrm{2}{a}_{\mathrm{3}} \:=\:\mathrm{60}\:−\:\mathrm{8}=\mathrm{52} \\ $$$${a}_{\mathrm{6}} \:=\:\mathrm{6}{a}_{\mathrm{5}} \:−\:\mathrm{2}{a}_{\mathrm{4}} \:=\:\mathrm{312}−\mathrm{24}=\mathrm{288} \\ $$