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Let-A-B-and-C-are-three-statments-A-B-C-A-C-B-A-B-C-A-B-A-Prove-or-disprove-

Question Number 1656 by Rasheed Soomro last updated on 29/Aug/15 $$\mathrm{Let}\:\mathrm{A},\mathrm{B}\:\mathrm{and}\:\mathrm{C}\:\mathrm{are}\:\mathrm{three}\:\mathrm{statments}.\: \\ $$$$\left(\mathrm{A}\Rightarrow\mathrm{B}\Rightarrow\mathrm{C}\Rightarrow\mathrm{A}\right)\:\overset{?} {\Rightarrow}\left(\mathrm{C}\Rightarrow\mathrm{B}\right)\: \\ $$$$\left(\mathrm{A}\Rightarrow\mathrm{B}\Rightarrow\mathrm{C}\Rightarrow\mathrm{A}\right)\:\overset{?} {\Rightarrow}\left(\mathrm{B}\Rightarrow\mathrm{A}\right)\: \\ $$$$\mathrm{Prove}\:\mathrm{or}\:\mathrm{disprove}. \\ $$ Commented by Yozzian last…

solve-inside-R-3-the-system-2x-y-z-1-x-2y-z-2-x-y-2z-3-

Question Number 67189 by mathmax by abdo last updated on 23/Aug/19 $${solve}\:{inside}\:{R}^{\mathrm{3}} \:{the}\:{system}\:\begin{cases}{\mathrm{2}{x}+{y}+{z}\:=\mathrm{1}}\\{{x}+\mathrm{2}{y}+{z}\:=\mathrm{2}}\end{cases} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left\{{x}+{y}+\mathrm{2}{z}\:=\mathrm{3}\right. \\ $$ Answered by MJS last updated on 23/Aug/19 $${D}=\begin{vmatrix}{\mathrm{2}}&{\mathrm{1}}&{\mathrm{1}}\\{\mathrm{1}}&{\mathrm{2}}&{\mathrm{1}}\\{\mathrm{1}}&{\mathrm{1}}&{\mathrm{2}}\end{vmatrix}=\mathrm{4}…

let-f-x-arctan-x-3-1-calculate-f-n-x-and-f-n-0-2-developp-f-at-integr-serie-3-calculate-0-1-arctan-x-3-dx-

Question Number 67187 by mathmax by abdo last updated on 23/Aug/19 $${let}\:{f}\left({x}\right)\:={arctan}\left({x}^{\mathrm{3}} \right) \\ $$$$\left.\mathrm{1}\right){calculate}\:{f}^{\left({n}\right)} \left({x}\right){and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{arctan}\left({x}^{\mathrm{3}} \right){dx} \\…

Calculate-I-a-b-0-1-t-a-1-t-b-dt-given-that-I-a-b-b-a-1-I-a-1-b-1-a-gt-0-b-gt-0-Use-the-fact-that-I-a-b-I-a-1-b-I-a-b-1-and-I-a-b-I-b-a-to-help-evaluate-I-a-b-

Question Number 1643 by 112358 last updated on 28/Aug/15 $${Calculate}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{I}\left({a},{b}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {t}^{{a}} \left(\mathrm{1}−{t}\right)^{{b}} {dt} \\ $$$${given}\:{that}\:{I}\left({a},{b}\right)=\frac{{b}}{{a}+\mathrm{1}}{I}\left({a}+\mathrm{1},{b}−\mathrm{1}\right) \\ $$$$\left({a}>\mathrm{0},{b}>\mathrm{0}\right).\:{Use}\:{the}\:{fact}\:{that} \\ $$$${I}\left({a},{b}\right)={I}\left({a}+\mathrm{1},{b}\right)+{I}\left({a},{b}+\mathrm{1}\right) \\ $$$${and}\:{I}\left({a},{b}\right)={I}\left({b},{a}\right)\: \\…

1-1-3-1-2-3-1-4-3-1-5-3-1-7-3-1-8-3-

Question Number 132715 by Dwaipayan Shikari last updated on 16/Feb/21 $$\frac{\mathrm{1}}{\mathrm{1}^{\mathrm{3}} }−\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{3}} }+\frac{\mathrm{1}}{\mathrm{4}^{\mathrm{3}} }−\frac{\mathrm{1}}{\mathrm{5}^{\mathrm{3}} }+\frac{\mathrm{1}}{\mathrm{7}^{\mathrm{3}} }−\frac{\mathrm{1}}{\mathrm{8}^{\mathrm{3}} }+… \\ $$ Answered by Olaf last updated on…

lets-x-gt-0-and-take-the-sequence-a-a-0-x-a-n-1-x-a-n-i-proof-that-0-a-n-a-n-1-ii-proof-that-M-such-that-a-n-M-iii-using-i-and-ii-proof-that-lim-n-a-n-exist-iv-compute-lim-n

Question Number 1635 by 123456 last updated on 28/Aug/15 $$\mathrm{lets}\:{x}>\mathrm{0},\:\mathrm{and}\:\mathrm{take}\:\mathrm{the}\:\mathrm{sequence}\:{a} \\ $$$${a}_{\mathrm{0}} =\sqrt{{x}} \\ $$$${a}_{{n}+\mathrm{1}} =\sqrt{{x}+{a}_{{n}} } \\ $$$$\mathrm{i}.\mathrm{proof}\:\mathrm{that}\:\mathrm{0}\leqslant{a}_{{n}} \leqslant{a}_{{n}+\mathrm{1}} \\ $$$$\mathrm{ii}.\mathrm{proof}\:\mathrm{that}\:\exists\mathrm{M}\:\mathrm{such}\:\mathrm{that}\:{a}_{{n}} \leqslant\mathrm{M} \\ $$$$\mathrm{iii}.\mathrm{using}\:\mathrm{i}\:\mathrm{and}\:\mathrm{ii}\:\mathrm{proof}\:\mathrm{that}\:\underset{{n}\rightarrow\infty}…

solve-for-real-x-and-y-a-b-R-a-x-3-1-y-3-x-2-1-y-2-b-x-3-x-2-1-y-3-x-2-x-1-y-2-c-x-3-y-2-9xy-x-2-y-3-8xy-d-ax-by-2ab-x-2-y-

Question Number 67167 by behi83417@gmail.com last updated on 23/Aug/19 $$\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{real}}\:\:\boldsymbol{\mathrm{x}}\:\boldsymbol{\mathrm{and}}\:\:\boldsymbol{\mathrm{y}}:\left[\mathrm{a},\mathrm{b}\in\mathrm{R}\right] \\ $$$$\boldsymbol{\mathrm{a}}.\begin{cases}{\boldsymbol{\mathrm{x}}^{\mathrm{3}} +\mathrm{1}=\boldsymbol{\mathrm{y}}^{\mathrm{3}} }\\{\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\mathrm{1}=\boldsymbol{\mathrm{y}}^{\mathrm{2}} }\end{cases}\:\:\:\:\:\:\:\: \\ $$$$\boldsymbol{\mathrm{b}}.\begin{cases}{\boldsymbol{\mathrm{x}}^{\mathrm{3}} +\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\mathrm{1}=\boldsymbol{\mathrm{y}}^{\mathrm{3}} }\\{\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{x}}+\mathrm{1}=\boldsymbol{\mathrm{y}}^{\mathrm{2}} }\end{cases} \\ $$$$\boldsymbol{\mathrm{c}}.\begin{cases}{\boldsymbol{\mathrm{x}}^{\mathrm{3}}…