Category: Algebra

Question Number 98215 by I want to learn more last updated on 12/Jun/20 $$\mathbf{Find}\mathbf{the}\mathbf{nth}\mathbf{term}\mathbf{of}\mathbf{the}\mathbf{sequence}\left\{{\mathbf{a}}_{\mathbf{n}}\right\}\mathbf{such}\mathbf{that}$$$$\frac{{\mathbf{a}}_1+{\mathbf{a}}_2+\dots +{\mathbf{a}}_{\mathbf{n}}}{\mathbf{n}}=\mathbf{n}+\frac{1}{\mathbf{n}}(\mathbf{n}=1,2,3,\dots )$$ Commented by Don08q…

Question Number 163741 by mathlove last updated on 10/Jan/22 Commented by Rasheed.Sindhi last updated on 10/Jan/22 $$Whattodo?$$ Commented by mathlove last updated on…

Question Number 98205 by I want to learn more last updated on 12/Jun/20 $$\mathbf{Find}\mathbf{the}\mathbf{nth}\mathbf{term}\mathbf{of}\mathbf{the}\mathbf{sequence}\left\{{\mathbf{a}}_{\mathbf{n}}\right\}\mathbf{such}\mathbf{that}$$$${\mathbf{a}}_1=1,{\mathbf{a}}_{\mathbf{n}+1}=\frac{1}{2}{\mathbf{a}}_{\mathbf{n}}+\frac{{\mathbf{n}}^2-2\mathbf{n}-1}{{\mathbf{n}}^2{(\mathbf{n}+1)}^2}(\mathbf{n}=1,2,3,\dots )$$…

Question Number 163736 by HongKing last updated on 09/Jan/22 Answered by mr W last updated on 10/Jan/22 Commented by mr W last updated on 10/Jan/22…

Question Number 163730 by HongKing last updated on 09/Jan/22 $$\mathrm{Find}:\mathbf{\Omega }=\int \frac{\sin \left(\mathrm{x}\right)+\sqrt{3}\cos \left(\mathrm{x}\right)}{\sin \left(3\mathrm{x}\right)}\mathrm{dx}$$ Answered by cortano1 last updated on 10/Jan/22 $$=2\int \frac{\sin (x+\frac{\pi }{3})}{\sin \left(3x\right)}dx$$$$=2\int \frac{d\left(\tan \theta \right)}{3-\tan {}^2\theta };\theta =x+\frac{\pi }{6}$$$$\:=\:\frac{\mathrm{2}\sqrt{\mathrm{3}}}{\mathrm{3}}\:\mathrm{tanh}\:^{−\mathrm{1}}…