Question Number 2240 by Rasheed Soomro last updated on 10/Nov/15 $$\mathcal{GENERALIZE}: \\ $$$$\left({a}+{b}\right)\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} −{ab}\right)={a}^{\mathrm{3}} +{b}^{\mathrm{3}} \\ $$$$\left({a}+{b}+{c}\right)\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{ab}−{bc}−{ca}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:={a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}}…
Question Number 67743 by Enock last updated on 31/Aug/19 $$\:^{} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 2207 by Yozzi last updated on 08/Nov/15 $${Find}\:{the}\:{sum}\:{to}\:{n}\:{terms}\:{of}\:{the}\:{series} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{S}\left({x}\right)=\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}{e}^{−{x}^{{r}} } . \\ $$ Commented by 123456 last updated on 09/Nov/15…
Question Number 133279 by greg_ed last updated on 20/Feb/21 $$\boldsymbol{\mathrm{hi}},\:\boldsymbol{\mathrm{everybody}}\:! \\ $$$$\boldsymbol{\mathrm{with}}\:\boldsymbol{\mathrm{n}}\:\in\:\mathbb{N}, \\ $$$$\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}}\::\:\exists\:\boldsymbol{\mathrm{n}}_{\mathrm{0}} \:\in\:\mathbb{N}\:/\:\forall\:\boldsymbol{\mathrm{n}}\:\geqslant\:\boldsymbol{\mathrm{n}}_{\mathrm{0}} \:,\:\boldsymbol{\mathrm{n}}^{\mathrm{2}} \:\leqslant\:\mathrm{2}^{\boldsymbol{\mathrm{n}}} . \\ $$ Terms of Service Privacy Policy…
Question Number 67711 by naka3546 last updated on 30/Aug/19 $${Find}\:\:{the}\:\:{value}\:\:{of} \\ $$$$\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{cos}^{\mathrm{2}} \left(\mathrm{10}°\right)}\:+\:\frac{\mathrm{1}}{\mathrm{sin}^{\mathrm{2}} \left(\mathrm{20}°\right)}\:+\:\frac{\mathrm{1}}{\mathrm{sin}^{\mathrm{2}} \left(\mathrm{40}°\right)}\: \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 2157 by Yozzi last updated on 05/Nov/15 $${Evaluate}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}+\frac{\mathrm{2}^{\mathrm{3}} }{\mathrm{1}!}+\frac{\mathrm{3}^{\mathrm{3}} }{\mathrm{2}!}+\frac{\mathrm{4}^{\mathrm{3}} }{\mathrm{3}!}+… \\ $$$${by}\:{considering}\:{the}\:{series}\:{expansion} \\ $$$${of}\:{an}\:{expression}\:{of}\:{the}\:{form}\:{P}\left({x}\right){e}^{{x}} \\ $$$${where}\:{P}\left({x}\right)\:{is}\:{a}\:{suitably}\:{chosen} \\ $$$${polynomial}\:{in}\:{x}.\: \\ $$$$…
Question Number 2135 by Rasheed Soomro last updated on 04/Nov/15 $${Solve}\:{the}\:{following}\:{system}\:{of}\:{inequalities} \\ $$$${b}^{\mathrm{2}} {x}^{\mathrm{2}} +{a}^{\mathrm{2}} {y}^{\mathrm{2}} \:\leqslant{a}^{\mathrm{2}} {b}^{\mathrm{2}} \:\:\wedge\:\:\:{a}^{\mathrm{2}} {x}^{\mathrm{2}} +{b}^{\mathrm{2}} {y}^{\mathrm{2}} \:\leqslant{a}^{\mathrm{2}} {b}^{\mathrm{2}\:} \:\:;\:\:\:{a},{b}\neq\mathrm{0}…
Question Number 2134 by Rasheed Soomro last updated on 04/Nov/15 $${Factorize} \\ $$$$−\mathrm{2}+\frac{\mathrm{1}}{{x}^{\mathrm{3}} }+{x}^{\mathrm{3}} \\ $$$$\left({Stepwise}\:{process}\:{is}\:{required}\right) \\ $$ Answered by sudhanshur last updated on 04/Nov/15…
Question Number 133201 by Tojiboyeva Kamolahon last updated on 19/Feb/21 $$\sqrt{\mathrm{81}} \\ $$ Answered by MJS_new last updated on 20/Feb/21 $$−\mathrm{3}^{\mathrm{2}} \mathrm{e}^{\mathrm{i}\pi} \\ $$ Commented…
Question Number 2112 by Yozzi last updated on 03/Nov/15 $${Prove}\:{that},\:\forall{n}\in\mathbb{N}, \\ $$$${H}\left(\mathrm{2}^{{n}} \right)\geqslant\mathrm{1}+\frac{{n}}{\mathrm{2}} \\ $$$${where}\:{H}\left({m}\right)=\underset{{r}=\mathrm{1}} {\overset{{m}} {\sum}}\frac{\mathrm{1}}{{r}}. \\ $$ Commented by 123456 last updated on…