Question Number 148229 by iloveisrael last updated on 26/Jul/21 Commented by nimnim last updated on 26/Jul/21 $$\:\:\:\:\:\:\:\:\:{x}+\sqrt{{x}^{\mathrm{2}} −\mathrm{1}}+\frac{\mathrm{1}}{{x}−\sqrt{{x}^{\mathrm{2}} −\mathrm{1}}}=\mathrm{20} \\ $$$$\Rightarrow\frac{{x}^{\mathrm{2}} −\left({x}^{\mathrm{2}} −\mathrm{1}\right)+\mathrm{1}}{{x}−\sqrt{{x}^{\mathrm{2}} −\mathrm{1}}}×\frac{{x}+\sqrt{{x}^{\mathrm{2}} −\mathrm{1}}}{{x}+\sqrt{{x}^{\mathrm{2}}…
Question Number 148226 by mathdanisur last updated on 26/Jul/21 Answered by mitica last updated on 26/Jul/21 $$\Sigma\frac{\mathrm{1}}{{x}\left({px}+\mathrm{1}\right)}=\Sigma\frac{\left(\frac{\mathrm{1}}{{x}}\right)^{\mathrm{2}} }{{p}+\frac{\mathrm{1}}{{x}}}\geqslant \\ $$$$\frac{\left(\frac{\mathrm{1}}{{x}}+\frac{\mathrm{1}}{{y}}+\frac{\mathrm{1}}{{z}}\right)^{\mathrm{2}} }{{p}+{q}+{r}+\frac{\mathrm{1}}{{x}}+\frac{\mathrm{1}}{{y}}+\frac{\mathrm{1}}{{z}}}=\frac{\mathrm{9}}{{p}+{q}+{r}+\mathrm{3}} \\ $$ Commented by…
Question Number 148203 by liberty last updated on 26/Jul/21 $$\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{8}}{{n}^{\mathrm{2}} +{n}}\right)=? \\ $$ Answered by mathmax by abdo last updated on 26/Jul/21 $$\mathrm{S}=\mathrm{8}\sum_{\mathrm{n}=\mathrm{1}}…
Question Number 82667 by M±th+et£s last updated on 23/Feb/20 $${if}\:{a}>\mathrm{0}\:\:{b}>\mathrm{0}\:\:{a}\leqslant{b} \\ $$$$ \\ $$$${show}\:{that}\: \\ $$$${a}^{\mathrm{2}} \leqslant\left(\frac{\mathrm{2}{ab}}{{a}+{b}}\right)^{\mathrm{2}} \leqslant{ab}\leqslant\left(\frac{{a}+{b}}{\mathrm{2}}\right)^{\mathrm{2}} \leqslant\frac{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }{\mathrm{2}}\leqslant{b}^{\mathrm{2}} \\ $$ Answered by…
Question Number 82665 by jagoll last updated on 23/Feb/20 $$\sqrt{\mathrm{6561}\sqrt{\mathrm{6561}\sqrt{\mathrm{6561}\sqrt{\mathrm{6561}\sqrt{…}}}}}\:=\:\mathrm{3}^{\:\mathrm{8}^{{x}} \:} \\ $$$$\sqrt{\mathrm{6561}\sqrt{…}}\:\:\left(\mathrm{60}\:{time}\right) \\ $$$${find}\:{x} \\ $$ Commented by mr W last updated on 23/Feb/20…
Question Number 82644 by Power last updated on 23/Feb/20 Commented by Power last updated on 23/Feb/20 $$\mathrm{sorry}\:\:\mathrm{it}\:\mathrm{worked} \\ $$ Commented by john santu last updated…
Question Number 148154 by mathdanisur last updated on 25/Jul/21 $$\sqrt{\mathrm{1}!\:+\:\mathrm{2}!\:+\:\mathrm{3}!\:+\:…\:+\:\boldsymbol{{n}}!}\:\:\in\:\mathbb{N} \\ $$$$\boldsymbol{{n}}\:=\:? \\ $$ Answered by puissant last updated on 25/Jul/21 $$\mathrm{n}=\mathrm{1} \\ $$ Commented…
Question Number 17065 by mrW1 last updated on 30/Jun/17 $$\mathrm{if}\:\mathrm{y}=\mathrm{x}^{\mathrm{x}} \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:\mathrm{this}\:\mathrm{function}? \\ $$ Commented by ajfour last updated on 30/Jun/17 $$\mathrm{y}\in\left[\mathrm{e}^{−\mathrm{1}/\mathrm{e}} ,\infty\right). \\ $$…
Question Number 82596 by peter frank last updated on 22/Feb/20 Commented by mr W last updated on 23/Feb/20 $$\left({a}\right) \\ $$$${i}\:{think}\:{the}\:{beam}\:{will}\:{stay}\:{in}\:{water}. \\ $$$$\left({b}\right) \\ $$$${the}\:{beam}\:{will}\:{be}\:{partially}\:{reflected}…
Question Number 82593 by Power last updated on 22/Feb/20 Commented by mr W last updated on 23/Feb/20 $${t}=\sqrt{{x}}>\mathrm{0} \\ $$$${t}^{\mathrm{4}} −\mathrm{16}{t}−\mathrm{12}=\mathrm{0} \\ $$$$\left({t}^{\mathrm{2}} −\mathrm{2}{t}−\mathrm{2}\right)\left({t}^{\mathrm{2}} +\mathrm{2}{t}+\mathrm{6}\right)=\mathrm{0}…