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Category: Algebra

if-we-let-f-n-n-2-n-1-then-solve-the-equation-for-real-numbers-f-f-f-f-n-1-32-

Question Number 154675 by mathdanisur last updated on 20/Sep/21 $$\mathrm{if}\:\mathrm{we}\:\mathrm{let} \\ $$$$\mathrm{f}\left(\mathrm{n}\right)\:=\:\frac{\mathrm{n}}{\mathrm{2}}\:\left(\mathrm{n}\:+\:\mathrm{1}\right) \\ $$$$\mathrm{then}\:\mathrm{solve}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers} \\ $$$$\mathrm{f}\left(\mathrm{f}\left(\mathrm{f}\left(\mathrm{f}\left(\mathrm{n}\right)\right)\right)\right)\:=\:\frac{\mathrm{1}}{\mathrm{32}} \\ $$ Answered by MJS_new last updated on 20/Sep/21…

if-n-N-gt-2-prove-that-n-1-3-n-2-1-3-3-1-0-mod-8-

Question Number 154649 by mathdanisur last updated on 20/Sep/21 $$\mathrm{if}\:\:\mathrm{n}\:\in\:\mathbb{N}^{>\mathrm{2}} \:\:\mathrm{prove}\:\mathrm{that} \\ $$$$\left[\left(\sqrt[{\mathrm{3}}]{\mathrm{n}}\:+\:\sqrt[{\mathrm{3}}]{\mathrm{n}\:+\:\mathrm{2}}\:\right)^{\mathrm{3}} \right]\:+\:\mathrm{1}\:=\:\mathrm{0}\:\left(\mathrm{mod}\:\mathrm{8}\right) \\ $$ Commented by MJS_new last updated on 20/Sep/21 $$\mathrm{just}\:\mathrm{a}\:\mathrm{try} \\…

let-be-A-1-1-0-1-B-1-0-1-1-find-e-A-e-B-1-e-A-exponential-matrix-

Question Number 154651 by mathdanisur last updated on 20/Sep/21 $$\mathrm{let}\:\mathrm{be}\:\:\boldsymbol{\mathrm{A}}\:=\:\begin{pmatrix}{\mathrm{1}}&{\mathrm{1}}\\{\mathrm{0}}&{\mathrm{1}}\end{pmatrix}\:\:;\:\:\boldsymbol{\mathrm{B}}\:=\:\begin{pmatrix}{\mathrm{1}}&{\mathrm{0}}\\{\mathrm{1}}&{\mathrm{1}}\end{pmatrix} \\ $$$$\mathrm{find}\:\:\boldsymbol{\Omega}\:=\:\mathrm{e}^{\boldsymbol{\mathrm{A}}} \:\centerdot\:\left(\mathrm{e}^{\boldsymbol{\mathrm{B}}} \right)^{−\mathrm{1}} \\ $$$$\left(\mathrm{e}^{\boldsymbol{\mathrm{A}}} \:-\:\mathrm{exponential}\:\mathrm{matrix}\right) \\ $$ Answered by TheHoneyCat last updated on…

f-Q-Q-f-x-f-y-y-f-x-x-y-Q-

Question Number 154647 by mathdanisur last updated on 20/Sep/21 $$\mathrm{f}\::\:\mathrm{Q}\:\rightarrow\:\mathrm{Q} \\ $$$$\mathrm{f}\left(\mathrm{x}\:+\:\mathrm{f}\left(\mathrm{y}\right)\right)\:=\:\mathrm{y}\:+\:\mathrm{f}\left(\mathrm{x}\right) \\ $$$$\forall\:\mathrm{x};\mathrm{y}\:\in\:\mathrm{Q} \\ $$ Answered by TheHoneyCat last updated on 20/Sep/21 $$\mathrm{I}\:\mathrm{demote}\:\mathrm{by}\:{f}^{{n}} \::\:{f}\circ…\circ{f}\:{n}\:\mathrm{times}…