Question Number 212075 by Spillover last updated on 28/Sep/24 Answered by Ghisom last updated on 28/Sep/24 $$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\sqrt{{n}+\sqrt{{n}}}−\sqrt{{n}}\right)\:=\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{1}+\sqrt{{t}}}−\mathrm{1}}{\:\sqrt{{t}}}\:= \\ $$$$\:\:\:\:\:\left[\mathrm{l}'\mathrm{H}\hat {\mathrm{o}pital}\right] \\ $$$$=\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{1}+\sqrt{{t}}}}\:=\frac{\mathrm{1}}{\mathrm{2}}…
Question Number 212066 by Spillover last updated on 28/Sep/24 Answered by Rasheed.Sindhi last updated on 29/Sep/24 $$\frac{\mathrm{2021}!+\mathrm{2020}!}{\mathrm{2021}!−\mathrm{2020}!}\:\centerdot\:\frac{\mathrm{2020}!+\mathrm{2019}!}{\mathrm{2020}!−\mathrm{2019}!}\:\centerdot…\frac{\mathrm{3}!+\mathrm{2}!}{\mathrm{3}!−\mathrm{2}!}\:\centerdot\:\frac{\mathrm{2}!+\mathrm{1}!}{\mathrm{2}!−\mathrm{1}!} \\ $$$$\frac{\left\{\left({x}+\mathrm{1}\right)!+{x}!\right\}}{\left\{\left({x}+\mathrm{1}\right)!−{x}!\right\}}=\frac{{x}!\left\{\left({x}+\mathrm{1}\right)+\mathrm{1}\right\}}{{x}!\left\{\left({x}+\mathrm{1}\right)−\mathrm{1}\right\}}=\frac{{x}+\mathrm{2}}{{x}} \\ $$$$\frac{\mathrm{2020}+\mathrm{2}}{\mathrm{2020}}\centerdot\frac{\mathrm{2019}+\mathrm{2}}{\mathrm{2019}}\centerdot…\frac{\mathrm{2}+\mathrm{2}}{\mathrm{2}}\centerdot\frac{\mathrm{1}+\mathrm{2}}{\mathrm{1}} \\ $$$$\frac{\left(\mathrm{2020}+\mathrm{2}\right)\left(\mathrm{2019}+\mathrm{2}\right)\left(\mathrm{2018}+\mathrm{2}\right)…\left(\mathrm{2}+\mathrm{2}\right)\left(\mathrm{1}+\mathrm{2}\right)}{\mathrm{2020}.\mathrm{2019}.\mathrm{2018}….\mathrm{2}.\mathrm{1}} \\ $$$$=\frac{\mathrm{2022}.\mathrm{2021}.\mathrm{2020}….\mathrm{4}.\mathrm{3}}{\mathrm{2020}!}…
Question Number 212065 by Spillover last updated on 28/Sep/24 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 212043 by RojaTaniya last updated on 28/Sep/24 $$\:{HCF}\:{of}\:\left\{\left({n}^{\mathrm{2}} +\mathrm{10}\right),\:\left({n}+\mathrm{1}\right)^{\mathrm{2}} +\mathrm{10}\right\}=? \\ $$$$\:\:{n}\in{N} \\ $$ Answered by A5T last updated on 28/Sep/24 $${Let}\:{a}={n}^{\mathrm{2}} +\mathrm{10},\:{b}=\left({n}+\mathrm{1}\right)^{\mathrm{2}}…
Question Number 212067 by Spillover last updated on 28/Sep/24 Answered by Spillover last updated on 28/Sep/24 Commented by Ghisom last updated on 28/Sep/24 $$\mathrm{that}'\mathrm{s}\:\mathrm{not}\:\mathrm{true}.\:\mathrm{we}\:\mathrm{can}\:\mathrm{solve}\:\mathrm{the}\:\mathrm{equation} \\…
Question Number 212024 by Spillover last updated on 27/Sep/24 Answered by Frix last updated on 27/Sep/24 $$\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{4}}} {\int}}\left(\mathrm{tan}\:{x}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} {dx}\:\overset{\left[{t}=\left(\mathrm{tan}\:{x}\right)^{−\frac{\mathrm{1}}{\mathrm{3}}} \right]} {=}\:\mathrm{3}\underset{\mathrm{1}} {\overset{\infty} {\int}}\frac{{dt}}{{t}^{\mathrm{6}} +\mathrm{1}}…
Question Number 212033 by RojaTaniya last updated on 27/Sep/24 $$\:{Find}\:{last}\:{two}\:{digit}\:{of}\:\mid\mathrm{33}^{\mathrm{22}} −\mathrm{22}^{\mathrm{33}} \mid \\ $$ Answered by Frix last updated on 27/Sep/24 $$\mathrm{22}^{\mathrm{33}} −\mathrm{33}^{\mathrm{22}} = \\…
Question Number 212016 by Spillover last updated on 27/Sep/24 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 211989 by Spillover last updated on 26/Sep/24 Answered by som(math1967) last updated on 26/Sep/24 $$\:{cos}\alpha{cos}\beta+{sin}\alpha{sin}\beta+{cos}\beta{cos}\gamma \\ $$$$+{sin}\beta{sin}\gamma+{cos}\gamma{cos}\alpha+{sin}\gamma{sin}\alpha \\ $$$$=−\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\mathrm{2}\left({cos}\alpha{cos}\beta+{cos}\beta{cos}\gamma+{cos}\gamma{cos}\alpha\right) \\ $$$$+\mathrm{2}\left({sin}\alpha{sin}\beta+{sin}\beta{sin}\gamma+{sin}\alpha{sin}\gamma\right)…
Question Number 211987 by Spillover last updated on 26/Sep/24 Commented by MathematicalUser2357 last updated on 26/Sep/24 $$\mathrm{No}\:\mathrm{closed}\:\mathrm{forms}\:\mathrm{for}\:\int_{\mathrm{0}} ^{\pi/\mathrm{2}} {x}^{\mathrm{2}} \left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{tan}\:{x}}}−\sqrt{\mathrm{tan}\:{x}}\right){dx}. \\ $$$$\mathrm{But}\:\mathrm{you}\:\mathrm{can}\:\mathrm{just}\:\mathrm{approximate}. \\ $$ Commented…