Question Number 138383 by henderson last updated on 12/Apr/21 $$\boldsymbol{\mathrm{hi}}\:! \\ $$$$\boldsymbol{\mathrm{for}}\:{a}_{\mathrm{0}} \:=\:\mathrm{1}\:\boldsymbol{\mathrm{and}}\:\forall\:{n}\:\geqslant\:\mathrm{1},\:{a}_{{n}} \:=\:\frac{\mathrm{1}}{{n}}\:\underset{{k}=\mathrm{0}} {\overset{\mathrm{n}−\mathrm{1}} {\sum}}\:\:\frac{{a}_{{k}} }{{n}−{k}}\:. \\ $$$$\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}}\:\forall\:{n}\:\geqslant\:\mathrm{0},\:\boldsymbol{\mathrm{we}}\:\boldsymbol{\mathrm{get}}\:\mathrm{0}\:\leqslant\:{a}_{{n}} \:\leqslant\:\mathrm{1}. \\ $$ Commented by mitica…
Question Number 7299 by Tawakalitu. last updated on 22/Aug/16 Commented by sou1618 last updated on 22/Aug/16 $${f}\left({x}\right)={x}^{\mathrm{4}} +\mathrm{2}{x}^{\mathrm{3}} −{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1} \\ $$$${x}={y}−\frac{\mathrm{2}}{\mathrm{4}} \\ $$$${f}\left({y}−\frac{\mathrm{1}}{\mathrm{2}}\right)=\left({y}−\mathrm{1}/\mathrm{2}\right)^{\mathrm{4}} +\mathrm{2}\left({y}−\mathrm{1}/\mathrm{2}\right)^{\mathrm{3}}…
Question Number 7264 by WagST last updated on 19/Aug/16 $${many}\:{thanks}. \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 138329 by JulioCesar last updated on 12/Apr/21 Answered by Ñï= last updated on 12/Apr/21 $$\int\frac{{dx}}{\mathrm{sin}\:^{\mathrm{4}} {x}+\mathrm{cos}\:^{\mathrm{4}} {x}}=\int\frac{{dx}}{\mathrm{1}−\mathrm{2sin}\:^{\mathrm{2}} {x}\mathrm{cos}\:^{\mathrm{2}} {x}}=\int\frac{{dx}}{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin}^{\mathrm{2}} \:\mathrm{2}{x}} \\ $$$$=\int\frac{{csc}^{\mathrm{2}} \mathrm{2}{xd}}{{csc}^{\mathrm{2}}…
Question Number 138330 by JulioCesar last updated on 12/Apr/21 Answered by Dwaipayan Shikari last updated on 12/Apr/21 $${tanh}^{−\mathrm{1}} {a}=\frac{\mathrm{1}}{\mathrm{2}}{log}\left(\frac{\mathrm{1}+{a}}{\mathrm{1}−{a}}\right) \\ $$$${tanh}^{−\mathrm{1}} {a}+{tanh}^{−\mathrm{1}} {b}=\frac{\mathrm{1}}{\mathrm{2}}{log}\left(\frac{\left(\mathrm{1}+{a}\right)\left(\mathrm{1}+{b}\right)}{\left(\mathrm{1}−{a}\right)\left(\mathrm{1}−{b}\right)}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}{log}\left(\frac{\mathrm{1}+{ab}+{a}+{b}}{\mathrm{1}+{ab}−{a}−{b}}\right)…
Question Number 7251 by Tawakalitu. last updated on 19/Aug/16 $${Find}\:{the}\:{value}\:{of}\:{x}\: \\ $$$$ \\ $$$$\left(\sqrt{\mathrm{2}\:+\:\sqrt{\mathrm{3}}}\right)^{{x}} \:\:+\:\:\left(\sqrt{\mathrm{2}\:−\:\sqrt{\mathrm{3}}}\right)^{{x}} \:\:=\:\:\mathrm{4} \\ $$ Commented by sou1618 last updated on 19/Aug/16…
Question Number 7234 by WagST last updated on 17/Aug/16 $${need}\:{help}\:{with}\:{step}\:{by}\:{step}.\:{thanks}. \\ $$$$ \\ $$$${x}\left({x}+\mathrm{4}\right)\left({x}−\mathrm{1}\right)=\mathrm{2}{x}\left({x}+\mathrm{4}\right) \\ $$$$ \\ $$$$ \\ $$ Answered by Yozzia last updated…
Question Number 7222 by Tawakalitu. last updated on 16/Aug/16 $${What}\:{are}\:{the}\:{possible}\:{solution}\:{satisfying} \\ $$$${x}^{{y}} \:=\:{y}^{{x}} \\ $$ Commented by Tawakalitu. last updated on 17/Aug/16 $${I}\:{appreciate}\:{your}\:{effort}.\:{thanks}\:{sir}. \\ $$…
Question Number 7207 by Tawakalitu. last updated on 16/Aug/16 $${x}^{\left(\mathrm{2}{x}/{y}\right)} \:\:×\:\:\:{y}^{\left({y}/{x}\right)} \:\:=\:\:\mathrm{4}\:\:\:\:\:\:………….\:\left({i}\right) \\ $$$${xy}^{\left({xy}\:+\:{yx}\right)} \:\:=\:\:\mathrm{16}\:\:\:\:\:…………\:\left({ii}\right) \\ $$$$ \\ $$$${Find}\:{the}\:{value}\:{of}\:{x}\:{and}\:{y} \\ $$ Commented by Yozzia last…
Question Number 7193 by peter james last updated on 15/Aug/16 Commented by Yozzia last updated on 15/Aug/16 $${S}\left({n}\right)=\mathrm{1}+\mathrm{2}^{\mathrm{2}} −\mathrm{3}^{\mathrm{3}} +\mathrm{4}+\mathrm{5}^{\mathrm{2}} −\mathrm{6}^{\mathrm{3}} +\mathrm{7}+\mathrm{8}^{\mathrm{2}} −\mathrm{9}^{\mathrm{3}} +… \\…