Question Number 150156 by Lekhraj last updated on 09/Aug/21
Answered by Ar Brandon last updated on 10/Aug/21
$${u}_{{n}} =−\mathrm{4},\:\mathrm{1},\:\mathrm{12},\:\mathrm{29},\:\mathrm{52},\:\mathrm{81},\:\mathrm{116},...\:{u}_{\mathrm{1}} =−\mathrm{4} \\ $$$$\Delta{u}_{{n}} =\mathrm{5},\:\mathrm{11},\:\mathrm{17},\:\mathrm{23},\:\mathrm{29},...\:\:{d}_{\mathrm{1}} =\mathrm{5} \\ $$$$\Delta^{\mathrm{2}} {u}_{{n}} =\mathrm{6},\:\mathrm{6},\:\mathrm{6},\:\mathrm{6},...\:\:\:{d}_{\mathrm{2}} =\mathrm{6} \\ $$$$\Delta^{\mathrm{3}} {u}_{{n}} =\mathrm{0},\:\mathrm{0},\:\mathrm{0},...\:\:\:{d}_{\mathrm{3}} =\mathrm{0} \\ $$$${u}_{{n}} ={u}_{\mathrm{1}} +\frac{{d}_{\mathrm{1}} \left({n}−\mathrm{1}\right)}{\mathrm{1}!}+\frac{{d}_{\mathrm{2}} \left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)}{\mathrm{2}!} \\ $$$$\:\:\:\:\:=−\mathrm{4}+\mathrm{5}\left({n}−\mathrm{1}\right)+\mathrm{3}\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right) \\ $$$$\:\:\:\:\:=−\mathrm{4}+\mathrm{5}{n}−\mathrm{5}+\mathrm{3}{n}^{\mathrm{2}} −\mathrm{9}{n}+\mathrm{6} \\ $$$$\:\:\:\:\:=\mathrm{3}{n}^{\mathrm{2}} −\mathrm{4}{n}−\mathrm{3} \\ $$
Commented by Lekhraj last updated on 10/Aug/21
$$\mathrm{Thank}\:\mathrm{you} \\ $$
Answered by JDamian last updated on 09/Aug/21
$$−\mathrm{4}\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{12}\:\:\:\:\mathrm{29}\:\:\:\:\:\mathrm{52}\:\:\:\:\:\:\mathrm{81}\:\:\:\:\:\:\mathrm{116}\:\:\:\:\:\:\mathrm{157}\: \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{5}\:\:\:\:\:\:\mathrm{11}\:\:\:\:\mathrm{17}\:\:\:\:\:\mathrm{23}\:\:\:\:\:\:\mathrm{29}\:\:\:\:\:\:\mathrm{35}\:\:\:\:\:\:\:\mathrm{41} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{6}\:\:\:\:\:\:\:\:\mathrm{6}\:\:\:\:\:\:\:\mathrm{6}\:\:\:\:\:\:\:\:\mathrm{6}\:\:\:\:\:\:\:\:\:\mathrm{6}\:\:\:\:\:\:\:\:\mathrm{6} \\ $$
Commented by Lekhraj last updated on 09/Aug/21
$$\mathrm{Thank}\:\mathrm{you}.\: \\ $$