Question Number 200284 by depressiveshrek last updated on 16/Nov/23 $$\mathrm{Prove}\:\mathrm{that}\:\mathrm{for}\:\mathrm{any}\:\mathrm{set}\:{A}\:\mathrm{containing}\:{n} \\ $$$$\mathrm{elements},\:\mid\mathcal{P}\left({A}\right)\mid=\mathrm{2}^{{n}} . \\ $$ Answered by AST last updated on 16/Nov/23 $${C}_{\mathrm{0}} ^{{n}} +^{{n}}…
Question Number 199624 by pascal889 last updated on 06/Nov/23 Commented by mr W last updated on 06/Nov/23 $${you}\:{have}\:{posted}\:{the}\:{same}\:{question} \\ $$$${one}\:{time}\:{and}\:{a}\:{comment}\:{was}\:{given}. \\ $$$${why}\:{don}'{t}\:{you}\:{check}\:{if}\:{your}\:{question} \\ $$$${is}\:{wrong}?\:{maybe}\:{the}\:{question}\:{is} \\…
Question Number 198550 by HomeAlone last updated on 21/Oct/23 Commented by HomeAlone last updated on 21/Oct/23 $${also}\:{C}_{\mathrm{2}} \:{using}\:{F}\:{and}\:\mathrm{oxygen}\:\boldsymbol{\mathrm{gx}}_{\mathrm{2}} \\ $$ Terms of Service Privacy Policy…
Question Number 198380 by pascal889 last updated on 18/Oct/23 Commented by Rasheed.Sindhi last updated on 18/Oct/23 $$\mathbb{U}=? \\ $$$${Universal}\:{set}\:{is}\:{not}\:{given} \\ $$ Commented by AST last…
Question Number 196950 by Erico last updated on 05/Sep/23 $$\mathrm{Prove}\:\mathrm{that}\:\underset{\:\mathrm{0}} {\int}^{\:\frac{\pi}{\mathrm{2}}} \frac{\mathrm{ln}\left(\mathrm{1}+\alpha\mathrm{sin}{t}\right)}{\mathrm{sin}{t}}{dt}=\:\frac{\pi^{\mathrm{2}} }{\mathrm{8}}−\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{arccos}\alpha\right)^{\mathrm{2}} \\ $$ Answered by Mathspace last updated on 06/Sep/23 $${f}\left({x}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{ln}\left(\mathrm{1}+{xsint}\right)}{{sint}}{dt}…
Question Number 195393 by Erico last updated on 01/Aug/23 $$\mathrm{prove}\:\mathrm{that}\: \\ $$$$\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\sqrt[{\:\:\boldsymbol{{x}}\:\:}]{\frac{\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}{k}}\right)^{{x}} }{{n}}}\:=\:\frac{\mathrm{1}}{\mathrm{4}}\sqrt[{\boldsymbol{{n}}}]{\mathrm{C}_{\mathrm{2}\boldsymbol{\mathrm{n}}} ^{\boldsymbol{\mathrm{n}}} } \\ $$ Answered by witcher3 last updated…
Question Number 195157 by Erico last updated on 25/Jul/23 $$\mathrm{Prove}\:\mathrm{that} \\ $$$$\frac{{x}^{\mathrm{3}} }{\mathrm{2}{sin}^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{2}}{arctan}\:\frac{{x}}{{y}}\right)}+\frac{{y}^{\mathrm{3}} }{\mathrm{2}{cos}^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{2}}{arctan}\:\frac{{y}}{{x}}\right)}=\left({x}+{y}\right)\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right) \\ $$ Answered by Frix last updated…
Question Number 194868 by Erico last updated on 17/Jul/23 $$\mathrm{Prove}\:\mathrm{that}\:\forall{n}\in\mathrm{IN} \\ $$$$\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} {t}\:{sin}^{\mathrm{2}{n}} \left({lnt}\right){dt}=\:\frac{\mathrm{1}}{\mathrm{1}−{e}^{−\mathrm{2}\pi} }\:\underset{\:\mathrm{0}} {\int}^{\:\pi} {e}^{−\mathrm{2}{t}} {sin}^{\mathrm{2}{n}} \left({t}\right){dt} \\ $$ Answered by witcher3…
Question Number 194781 by MM42 last updated on 15/Jul/23 $${f}_{{n}\:} \:\:{the}\:{general}\:{sentence}\:{is}\:{seqiencee} \\ $$$${fibonacci}.\: \\ $$$${prove}\:{that}\::\:\:{f}_{\mathrm{2}{n}−\mathrm{1}} ={f}_{{n}} ^{\mathrm{2}} +{f}_{{n}−\mathrm{1}} ^{\mathrm{2}} \\ $$$$ \\ $$ Terms of…
Question Number 194693 by MM42 last updated on 13/Jul/23 $${if}\:\:\:{f}_{{n}} ={f}_{{n}−\mathrm{1}} +{f}_{{n}−\mathrm{2}} \:\:;\:\:{f}_{\mathrm{1}} ={f}_{\mathrm{2}} =\mathrm{1} \\ $$$${then}\:\:\:{prove}\:{that}\:\:\:\mathrm{5}\mid{f}_{\mathrm{5}{n}} \:\: \\ $$ Answered by Frix last updated…