Question Number 195343 by Matica last updated on 31/Jul/23 $$\:{hello}\:{everyone},\:{I}'{m}\:{a}\:{new}\:{student}. \\ $$$$\:{I}\:{beg}\:{your}\:{pardon}. \\ $$$$\:{Please}\:{tell}\:{me}\:{any}\:{good}\:{math}\:{book} \\ $$$$\:{you}\:{know}.\:{Thanks}. \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 195342 by Matica last updated on 31/Jul/23 $$\:\:\mathrm{1}.\:\mathrm{Prove}\:\mathrm{that}\:\:\forall{n}\:\in\:\mathbb{N}^{\ast} \:,\:\mathrm{4}^{{n}} \left({n}!\right)^{\mathrm{3}} \:<\:\left({n}+\mathrm{1}\right)^{\mathrm{3}{n}} \:. \\ $$$$\mathrm{2}.\:\mathrm{Solve}\:\mathrm{the}\:\mathrm{equations}\:\mathrm{in}\:\mathbb{Z}^{\mathrm{2}} \:: \\ $$$$\:\:\:\:\:{a}./\:\:\mathrm{2}{x}^{\mathrm{3}} +{xy}−\mathrm{7}=\mathrm{0}\:, \\ $$$$\:\:\:\:\:{b}./\:\:{x}\left({x}+\mathrm{1}\right)\left({x}+\mathrm{7}\right)\left({x}+\mathrm{8}\right)={y}^{\mathrm{2}} . \\ $$…
Question Number 195369 by Shlock last updated on 31/Jul/23 Answered by witcher3 last updated on 01/Aug/23 $$\mathrm{x}=−\mathrm{y} \\ $$$$\Rightarrow\mathrm{f}\left(\mathrm{f}\left(\mathrm{0}\right)\right)=\mathrm{2f}\left(\mathrm{x}^{\mathrm{2}} \right) \\ $$$$\Rightarrow\frac{\mathrm{1}}{\mathrm{2}}\mathrm{f}\left(\mathrm{f}\left(\mathrm{0}\right)\right)=\mathrm{f}\left(\mathrm{x}^{\mathrm{2}} \right) \\ $$$$\forall\left(\mathrm{x},\mathrm{y}\right)\in\mathbb{Z}\:^{\mathrm{2}}…
Question Number 195368 by Mingma last updated on 31/Jul/23 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 195370 by enrikalarcon last updated on 31/Jul/23 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 195365 by Mr.D.N. last updated on 31/Jul/23 $$\:\: \\ $$$$\:\underset{\mathrm{x}\rightarrow\mathrm{y}} {\overset{\mathrm{lim}} {\:}}\:\frac{\mathrm{tan}\:\mathrm{x}−\mathrm{tany}}{\mathrm{x}−\mathrm{y}} \\ $$$$ \\ $$ Answered by cortano12 last updated on 31/Jul/23…
Question Number 195364 by Rodier97 last updated on 01/Aug/23 $$ \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{for}\:\mathrm{any}\:\mathrm{natural}\:\mathrm{number}\:{n},\: \\ $$$$\mathrm{the}\:\mathrm{natural}\:\mathrm{number}\:\left(\mathrm{3}−\sqrt{\mathrm{5}}\right)^{{n}} +\left(\mathrm{3}+\sqrt{\mathrm{5}}\right)^{{n}} \:\mathrm{is}\:\mathrm{divisible} \\ $$$$\mathrm{by}\:\mathrm{2}^{{n}} . \\ $$ Answered by Frix last…
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Question Number 195325 by mathlove last updated on 30/Jul/23 $${prove}\:{that} \\ $$$$\underset{{x}\rightarrow\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\:\frac{{tan}\left(\frac{{x}}{\mathrm{2}}\right)−\mathrm{1}}{{x}−\frac{\pi}{\mathrm{2}}}=\mathrm{1} \\ $$ Answered by BaliramKumar last updated on 30/Jul/23 $$\underset{{x}\rightarrow\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\:\frac{\frac{\mathrm{d}}{\mathrm{dx}}\left({tan}\left(\frac{{x}}{\mathrm{2}}\right)−\mathrm{1}\right)}{\frac{\mathrm{d}}{\mathrm{dx}}\left({x}−\frac{\pi}{\mathrm{2}}\right)}\:=\:\frac{\mathrm{sec}^{\mathrm{2}} \left(\frac{\mathrm{x}}{\mathrm{2}}\right)\centerdot\frac{\mathrm{1}}{\mathrm{2}}}{\mathrm{1}}…
Question Number 195320 by Erico last updated on 30/Jul/23 $$\mathrm{I}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\:+\infty} {t}^{−\mathrm{2}{t}} {sin}^{\mathrm{2}{n}} {tdt} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{I}_{\mathrm{n}} =\frac{\mathrm{1}}{\mathrm{1}−{e}^{−\mathrm{2}\pi} }\:\:\underset{\:\mathrm{0}} {\int}^{\:\pi} {e}^{−\mathrm{2}{t}} {sin}^{\mathrm{2}{n}} {t}\:{dt} \\ $$$$\mathrm{and}\:\:\mathrm{I}_{\mathrm{n}}…