Question Number 213796 by efronzo1 last updated on 17/Nov/24 Answered by A5T last updated on 17/Nov/24 $${x}_{\mathrm{0}} ={k}\Rightarrow{x}_{\mathrm{1}} =\frac{\mathrm{1}+{k}}{\mathrm{1}−{k}}\Rightarrow{x}_{\mathrm{2}} =\frac{−\mathrm{1}}{{k}}\Rightarrow{x}_{\mathrm{3}} =\frac{{k}−\mathrm{1}}{\mathrm{1}+{k}}\Rightarrow{x}_{\mathrm{4}} =\frac{\mathrm{2}{k}}{\mathrm{2}}={k} \\ $$$$\Rightarrow{x}_{\mathrm{4}{n}} ={k}=\mathrm{2022}…
Question Number 212686 by mr W last updated on 21/Oct/24 $${in}\:{how}\:{many}\:{ways}\:{can}\:{a}\:{teacher} \\ $$$${divide}\:{his}\:\mathrm{10}\:{studens}\:{into}\:\mathrm{4}\:{groups} \\ $$$${such}\:{that}\:{each}\:{group}\:{has}\:{at}\:{least}\:\mathrm{2}\: \\ $$$${students}? \\ $$ Commented by Spillover last updated on…
Question Number 210290 by Adeyemi889 last updated on 05/Aug/24 Answered by mr W last updated on 05/Aug/24 $$\left({i}\right) \\ $$$${C}_{\mathrm{3}} ^{\mathrm{7}} {C}_{\mathrm{3}} ^{\mathrm{6}} =\mathrm{700}\:{ways} \\…
Question Number 209281 by Tawa11 last updated on 06/Jul/24 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{r},\:\mathrm{if}\:\:\overset{\mathrm{10}} {\:}\mathrm{C}_{\mathrm{r}} \:\:=\:\:\overset{\mathrm{10}} {\:}\mathrm{C}_{\mathrm{2r}\:\:+\:\:\mathrm{1}} \\ $$ Commented by klipto last updated on 06/Jul/24 $$\:\:\:\:\:\:\:^{\boldsymbol{\mathrm{n}}} \boldsymbol{\mathrm{C}}_{\mathrm{r}} =^{\mathrm{n}}…
Question Number 208662 by efronzo1 last updated on 20/Jun/24 $$\:\:\frac{\begin{pmatrix}{\mathrm{n}}\\{\mathrm{0}}\end{pmatrix}\:+\mathrm{3}\begin{pmatrix}{\mathrm{n}}\\{\mathrm{1}}\end{pmatrix}\:+\mathrm{5}\begin{pmatrix}{\mathrm{n}}\\{\mathrm{2}}\end{pmatrix}\:+…+\left(\mathrm{2n}+\mathrm{1}\right)\begin{pmatrix}{\mathrm{n}}\\{\mathrm{n}}\end{pmatrix}}{\begin{pmatrix}{\mathrm{n}}\\{\mathrm{1}}\end{pmatrix}\:+\mathrm{2}\begin{pmatrix}{\mathrm{n}}\\{\mathrm{2}}\end{pmatrix}\:+\:\mathrm{3}\begin{pmatrix}{\mathrm{n}}\\{\mathrm{3}}\end{pmatrix}\:+…+\mathrm{n}\begin{pmatrix}{\mathrm{n}}\\{\mathrm{n}}\end{pmatrix}}\:=\frac{\mathrm{23}}{\mathrm{11}} \\ $$$$\:\mathrm{n}=? \\ $$ Answered by Berbere last updated on 20/Jun/24 $${A}=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left(\mathrm{2}{k}+\mathrm{1}\right)\begin{pmatrix}{{n}}\\{{k}}\end{pmatrix};\underset{{k}=\mathrm{0}} {\overset{{n}}…
Question Number 208263 by efronzo1 last updated on 09/Jun/24 Answered by Ghisom last updated on 09/Jun/24 $${a}=\mathrm{e}^{\alpha} \:\Rightarrow\:{b}=\mathrm{e}^{\lambda} {a}\wedge{c}=\mathrm{e}^{\mathrm{2}\lambda} {a} \\ $$$${A}=\mathrm{log}_{{c}} \:{a}\:=\frac{\alpha}{\alpha+\mathrm{2}\lambda} \\ $$$${B}=\mathrm{log}_{{b}}…
Question Number 208238 by alcohol last updated on 08/Jun/24 $$\mathrm{S}{how}\:{that} \\ $$$$\frac{\pi}{\mathrm{4}}\:<\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\sqrt{\mathrm{1}−{x}^{\mathrm{4}} }{dx}\:{using}\:{x}\:=\:{sint} \\ $$$${show}\:{that}\:\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}−{x}^{\mathrm{4}} }{dx}<\frac{\mathrm{2}\sqrt{\mathrm{2}}}{\mathrm{3}} \\ $$$${using}\:\left(\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){g}\left({x}\right){dx}\right)^{\mathrm{2}} <\int_{\mathrm{0}}…
Question Number 207937 by efronzo1 last updated on 31/May/24 $$\:\:\mathrm{An}\:\mathrm{ordered}\:\mathrm{data}\:\mathrm{consists}\:\mathrm{of}\: \\ $$$$\:\mathrm{6}\:\mathrm{even}\:\mathrm{numbers}\:\mathrm{and}\:\mathrm{4}\:\mathrm{odd}\:\mathrm{numbers}. \\ $$$$\:\mathrm{The}\:\mathrm{average}\:\mathrm{of}\:\mathrm{the}\:\mathrm{odd}\:\mathrm{numbers}\: \\ $$$$\:\mathrm{is}\:\mathrm{2022}.\:\mathrm{The}\:\mathrm{3rd},\:\mathrm{5th},\:\mathrm{6th}\:\mathrm{and} \\ $$$$\:\mathrm{8}\:\mathrm{th}\:\mathrm{numbers}\:\mathrm{are}\:\mathrm{odd}.\: \\ $$$$\:\mathrm{The}\:\mathrm{data}\:\mathrm{range}\:\mathrm{is}\:\mathrm{24}\:,\:\mathrm{and}\:\mathrm{the}\: \\ $$$$\:\mathrm{interquartile}\:\mathrm{range}\:\mathrm{is}\:\mathrm{14}.\: \\ $$$$\:\mathrm{The}\:\mathrm{largest}\:\mathrm{possible}\:\mathrm{average} \\…
Question Number 207179 by efronzo1 last updated on 08/May/24 Answered by mr W last updated on 08/May/24 $$\mathrm{3}×\mathrm{4}×\mathrm{3}×\mathrm{2}=\mathrm{72}\:{such}\:{numbers} \\ $$ Commented by efronzo1 last updated…
Question Number 206827 by Fabricista15 last updated on 27/Apr/24 $$\mathrm{log}\:\left({x}\right)\:=\:\mathrm{sin}\:\left({x}\right) \\ $$$${x}\:=\:? \\ $$ Commented by Ghisom last updated on 27/Apr/24 $$\mathrm{if}\:\mathrm{log}\:{x}\:=\mathrm{ln}\:{x}\:\Rightarrow \\ $$$$\:\:\:\:\:{x}\approx\mathrm{2}.\mathrm{21910715} \\…