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…
Question Number 133180 by mathlove last updated on 19/Feb/21 Commented by mr W last updated on 19/Feb/21 $${no}\:{real}\:{solution}! \\ $$$${x}^{{x}} \geqslant\frac{\mathrm{1}}{\:\sqrt[{{e}}]{{e}}}\approx\mathrm{0}.\mathrm{692} \\ $$$$\frac{\mathrm{1}}{\mathrm{256}}<<\mathrm{0}.\mathrm{692} \\ $$$$\Rightarrow{x}^{{x}}…
Question Number 67574 by pete last updated on 28/Aug/19 $$\mathrm{Solve}\:\mathrm{x}^{\mathrm{2}} +\mathrm{1}<−\mathrm{5} \\ $$ Commented by mathmax by abdo last updated on 28/Aug/19 $$\left({e}\right)\Rightarrow{x}^{\mathrm{2}} +\mathrm{6}<\mathrm{0}\:\:\:{impossible}\:{equation}\:\Rightarrow{no}\:{solution}\: \\…
Question Number 133072 by bemath last updated on 18/Feb/21 $$\mathrm{Show}\:\mathrm{that}\:\frac{\mathrm{5}}{\mathrm{2}−\sqrt[{\mathrm{4}}]{\mathrm{3}}}\:\mathrm{is}\:\mathrm{in}\:\mathrm{F}_{\mathrm{2}} \:\mathrm{by}\:\mathrm{expressing} \\ $$$$\mathrm{the}\:\mathrm{number}\:\mathrm{in}\:\mathrm{form}\:{a}_{\mathrm{1}} +{b}_{\mathrm{1}} \sqrt{{k}_{\mathrm{1}} }\:\mathrm{where} \\ $$$${a}_{\mathrm{1}} ,{b}_{\mathrm{1}} ,\:{k}_{\mathrm{1}} \:{are}\:{in}\:{F}_{\mathrm{1}} \\ $$ Answered by…
Question Number 2000 by Yozzi last updated on 29/Oct/15 $${Find}\:{a}\:{non}−{constant}\:{function}\:{f}\: \\ $$$${satisfying}\:{f}\left(\mathrm{0}\right)=\mathrm{1},{f}\left(−\mathrm{2}\right)=\mathrm{0}\:{and} \\ $$$${f}\left({x}−{y}\right)={f}\left({x}\right){f}\left({y}\right)−{f}\left(−\mathrm{2}−{x}\right){f}\left({y}−\mathrm{2}\right). \\ $$ Commented by prakash jain last updated on 29/Oct/15 $${x}=\mathrm{0}…
Question Number 1988 by Rasheed Soomro last updated on 28/Oct/15 $${x}^{\mathrm{2}} =\:\frac{{f}\left({x}\right)+{f}\left(−{x}\right)}{\mathrm{2}} \\ $$$${f}\left({x}\right)=? \\ $$ Answered by 123456 last updated on 28/Oct/15 $$\mathrm{supossing}\:\mathrm{that}\:{f}\:\mathrm{is}\:\mathrm{poly} \\…