Question Number 131158 by EDWIN88 last updated on 02/Feb/21
$${If}\:\mathrm{4}{a}_{{n}} +\mathrm{2}{a}_{−{n}} =\mathrm{3}{n}^{\mathrm{2}} +\mathrm{2}{n}−\mathrm{3} \\ $$$${find}\:{a}_{{n}} \:=? \\ $$
Answered by mr W last updated on 02/Feb/21
$$\mathrm{4}{a}_{{n}} +\mathrm{2}{a}_{−{n}} =\mathrm{3}{n}^{\mathrm{2}} +\mathrm{2}{n}−\mathrm{3} \\ $$$$\mathrm{4}{a}_{−{n}} +\mathrm{2}{a}_{{n}} =\mathrm{3}{n}^{\mathrm{2}} −\mathrm{2}{n}−\mathrm{3} \\ $$$$\mathrm{8}{a}_{{n}} +\mathrm{4}{a}_{−{n}} =\mathrm{6}{n}^{\mathrm{2}} +\mathrm{4}{n}−\mathrm{6} \\ $$$$\Rightarrow\mathrm{6}{a}_{{n}} =\mathrm{3}{n}^{\mathrm{2}} +\mathrm{6}{n}−\mathrm{3} \\ $$$$\Rightarrow{a}_{{n}} =\frac{\mathrm{1}}{\mathrm{2}}\left({n}^{\mathrm{2}} +\mathrm{2}{n}−\mathrm{1}\right) \\ $$