Question Number 146290 by SLVR last updated on 12/Jul/21
Commented by SLVR last updated on 12/Jul/21
$${Dear}\:{Olaf}\:{sir}\:{how}\:{r}\:{is}\:{taken}.. \\ $$$${how}\:{f}\left({x}\right)\:{was}\:{constructed}??? \\ $$$${kindly}\:{explain}... \\ $$$$ \\ $$
Commented by SLVR last updated on 12/Jul/21
$${kindy}\:{tell}\:{me}\:{how}\:{r}^{\mathrm{2}} {and}\:{r}\:{were}\: \\ $$$${related}\:{to}\:{u}_{{n}} ??{I}\:{hope}\:{Auxilary}\:{equation} \\ $$$${by}\:{u}_{{n}} ={f}\left({x}+{n}\right)={r}^{{n}} \:??? \\ $$
Commented by Olaf_Thorendsen last updated on 12/Jul/21
$$\mathrm{Let}\:{u}_{{n}} \:=\:{f}\left({x}+{n}\right) \\ $$$$\mathrm{then}\:{u}_{{n}+\mathrm{2}} −\mathrm{7}{u}_{{n}+\mathrm{1}} +\mathrm{10}\:=\:\mathrm{0} \\ $$$$\mathrm{You}\:\mathrm{demonstrate}\:\mathrm{then}\:\mathrm{that}\: \\ $$$${u}_{{n}} \:=\:{f}\left({n}+{x}\right)\:=\:\mathrm{2}^{{n}+{x}} +\mathrm{5}^{{n}+{x}} \:\mathrm{by}\:\mathrm{induction} \\ $$
Commented by Olaf_Thorendsen last updated on 12/Jul/21
$${see}\:{the}\:{chapter}\:{vector}\:{space}\:{and} \\ $$$${linear}\:{map} \\ $$
Commented by SLVR last updated on 13/Jul/21
$${Thanks}\:{a}\:{lotMr}.\:{Olaf} \\ $$