Question Number 95167 by Rio Michael last updated on 23/May/20 $$\mathrm{the}\:\mathrm{first}\:\mathrm{term}\:\mathrm{in}\:\mathrm{a}\:\mathrm{geometric}\:\mathrm{series}\:\mathrm{is}\:\frac{\left(\mathrm{2}{x}\:+\:\mathrm{7}\right)}{\mathrm{2}{x}−\mathrm{5}}\:\mathrm{and}\:\mathrm{the}\:\mathrm{common}\:\mathrm{ratio}\:\mathrm{is} \\ $$$$\:\frac{\left(\mathrm{2}{x}−\mathrm{5}\right)}{\mathrm{2}{x}\:+\:\mathrm{7}}\:\mathrm{find}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{values}\:\mathrm{of}\:{x}\:\mathrm{for}\:\mathrm{which}\:\mathrm{all}\:\mathrm{the}\:\mathrm{terms}\:\mathrm{are}\:\mathrm{possible}. \\ $$ Commented by john santu last updated on 23/May/20 $$\left(\mathrm{1}\right)\:\mathrm{x}\:\neq\:\frac{\mathrm{5}}{\mathrm{2}}\:\wedge\:\mathrm{x}\:\neq\:−\frac{\mathrm{7}}{\mathrm{2}}\: \\…
Question Number 95164 by mr W last updated on 23/May/20 $${if}\:{a}_{{k}} =\mathrm{tan}\:\left(\theta+\frac{{k}\pi}{{n}}\right), \\ $$$${find}\:\frac{\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{a}_{{k}} }{\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}{a}_{{k}} }=? \\ $$ Commented by MJS…
Question Number 95159 by i jagooll last updated on 23/May/20 $$\sqrt[{\mathrm{3}\:\:}]{\mathrm{8}−\mathrm{x}}\:+\:\sqrt{\mathrm{x}}\:=\:\mathrm{2}\: \\ $$ Commented by hknkrc46 last updated on 23/May/20 $$\bigstar\:\sqrt[{\mathrm{2}{n}+\mathrm{1}}]{{f}\left({x}\right)}\:\Rightarrow\:\forall{f}\left({x}\right)\:\in\:\mathbb{R}\:\rightarrow\:{f}\left({x}\right)\geqslant\mathrm{0}\:\vee\:{f}\left({x}\right)\leqslant\mathrm{0} \\ $$$$\bigstar\sqrt[{\mathrm{2}{n}}]{{f}\left({x}\right)}\:\Rightarrow\:\forall{f}\left({x}\right)\:\in\:\mathbb{R}^{+} \:\rightarrow\:{f}\left({x}\right)\geqslant\mathrm{0} \\…
Question Number 95158 by M±th+et+s last updated on 23/May/20 $${find}\:{the}\:{domain}\:{and}\:{range} \\ $$$${f}\left({x}\right)=\lfloor\frac{\mathrm{1}}{{sin}\left\{{x}\right\}}\rfloor \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
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Question Number 160677 by tounghoungko last updated on 04/Dec/21 $$\:\:\:\begin{cases}{\frac{\mathrm{x}+\mathrm{y}}{\mathrm{xyz}}\:=\:−\frac{\mathrm{1}}{\mathrm{4}}}\\{\frac{\mathrm{y}+\mathrm{z}}{\mathrm{xyz}}\:=\:−\frac{\mathrm{1}}{\mathrm{24}}}\\{\frac{\mathrm{x}+\mathrm{z}}{\mathrm{xyz}}\:=\:\frac{\mathrm{1}}{\mathrm{24}}}\end{cases}\: \\ $$$$\:\: \\ $$ Commented by bobhans last updated on 04/Dec/21 $$\Leftrightarrow\:\frac{\mathrm{1}}{\mathrm{xyz}}\left(\mathrm{x}+\mathrm{y}+\mathrm{2z}\right)=\mathrm{0}\:\Rightarrow\mathrm{x}+\mathrm{y}=−\mathrm{2z} \\ $$$$\Leftrightarrow\:\frac{−\mathrm{2z}}{\mathrm{xyz}}\:=\:−\frac{\mathrm{1}}{\mathrm{4}}\:;\:\mathrm{xy}=\mathrm{8} \\…
Question Number 95127 by unknown last updated on 23/May/20 Commented by unknown last updated on 23/May/20 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{solution}\:\left({n},{k}\right).\:\mathrm{If}\:\left({n},{k}\right)\:\mathrm{is}\:\mathrm{natural}\:\mathrm{number}. \\ $$ Terms of Service Privacy Policy Contact:…
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Question Number 160658 by HongKing last updated on 04/Dec/21 $$\boldsymbol{\mathrm{Find}}:\:\:\:\underset{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{4}\boldsymbol{\mathrm{x}}\:\left[\:\frac{\mathrm{1}}{\mathrm{4}\boldsymbol{\mathrm{x}}}\:\right]\:=\:? \\ $$$$ \\ $$ Commented by mr W last updated on 04/Dec/21 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}4}{x}\left[\frac{\mathrm{1}}{\mathrm{4}{x}}\right]…
Question Number 160644 by HongKing last updated on 03/Dec/21 $$\mathrm{King}'\mathrm{s}\:\mathrm{inequality} \\ $$$$\mathrm{if}\:\:\:\mathrm{a}\leqslant\mathrm{b}\:\:\:;\:\:\:\mathrm{n}\leqslant\mathrm{m}\:\:\:\mathrm{then}: \\ $$$$\frac{\mathrm{a}^{\boldsymbol{\mathrm{n}}} \:+\:\mathrm{b}^{\boldsymbol{\mathrm{m}}} }{\mathrm{2}}\:\geqslant\:\left(\frac{\mathrm{1}\:+\:\mathrm{b}}{\mathrm{2}}\right)^{\boldsymbol{\mathrm{m}}-\boldsymbol{\mathrm{n}}} \:\centerdot\:\left(\frac{\mathrm{a}\:+\:\mathrm{b}}{\mathrm{2}}\right)^{\boldsymbol{\mathrm{n}}} \\ $$ Terms of Service Privacy Policy Contact:…