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Category: Algebra

let-p-q-be-reals-such-that-p-gt-q-gt-0-define-the-sequence-x-n-where-x-1-p-q-and-x-n-x-1-pq-x-n-1-for-n-2-for-all-n-then-x-n-

Question Number 210868 by universe last updated on 20/Aug/24 $$\mathrm{let}\:\mathrm{p}\:,\mathrm{q}\:\mathrm{be}\:\mathrm{reals}\:\mathrm{such}\:\mathrm{that}\:\mathrm{p}>\mathrm{q}>\mathrm{0}\:\mathrm{define} \\ $$$$\mathrm{the}\:\mathrm{sequence}\:\left\{\mathrm{x}_{\mathrm{n}} \right\}\:\mathrm{where}\:\mathrm{x}_{\mathrm{1}} =\:\mathrm{p}+\mathrm{q}\:\mathrm{and} \\ $$$$\mathrm{x}_{\mathrm{n}} \:=\:\mathrm{x}_{\mathrm{1}} −\frac{\mathrm{pq}}{\mathrm{x}_{\mathrm{n}−\mathrm{1}} }\:\mathrm{for}\:\mathrm{n}\geqslant\mathrm{2}\:\mathrm{for}\:\mathrm{all}\:\mathrm{n}\:\mathrm{then}\:\mathrm{x}_{\mathrm{n}} \:=\:?? \\ $$ Commented by Ghisom…

7-x-2-10-x-7-x-1-3-15x-6-0-x-

Question Number 210871 by hardmath last updated on 20/Aug/24 $$\mathrm{7}\left(\mathrm{x}−\mathrm{2}\right)\left(\sqrt{\mathrm{10}\:+\:\mathrm{x}}\:−\:\sqrt[{\mathrm{3}}]{\mathrm{7}−\mathrm{x}}\right)−\mathrm{15x}\:+\:\mathrm{6}\:=\:\mathrm{0} \\ $$$$\boldsymbol{\mathrm{x}}\:=\:? \\ $$ Commented by Ghisom last updated on 20/Aug/24 $$\mathrm{obviously}\:{x}=−\mathrm{1} \\ $$ Terms…

8x-2-13x-11-2-x-1-3-x-3x-2-2-1-3-x-

Question Number 210870 by hardmath last updated on 20/Aug/24 $$\mathrm{8x}^{\mathrm{2}} \:−\:\mathrm{13x}\:+\:\mathrm{11}\:=\:\frac{\mathrm{2}}{\mathrm{x}}\:+\:\left(\mathrm{1}\:+\:\frac{\mathrm{3}}{\mathrm{x}}\right)\:\sqrt[{\mathrm{3}}]{\mathrm{3x}^{\mathrm{2}} −\mathrm{2}} \\ $$$$\boldsymbol{\mathrm{x}}\:=\:? \\ $$ Commented by Ghisom last updated on 20/Aug/24 $$\mathrm{obviously}\:{x}=\mathrm{1} \\…

Solve-The-Equation-7x-1-9x-1-21x-1-63x-1-160-189-

Question Number 210858 by hardmath last updated on 20/Aug/24 $$\mathrm{Solve}\:\mathrm{The}\:\mathrm{Equation}: \\ $$$$\left(\mathrm{7x}+\mathrm{1}\right)\left(\mathrm{9x}+\mathrm{1}\right)\left(\mathrm{21x}+\mathrm{1}\right)\left(\mathrm{63x}+\mathrm{1}\right)=\:\frac{\mathrm{160}}{\mathrm{189}} \\ $$ Answered by mm1342 last updated on 20/Aug/24 $$\left(\mathrm{63}{x}+\mathrm{9}\right)\left(\mathrm{63}{x}+\mathrm{7}\right)\left(\mathrm{63}{x}+\mathrm{3}\right)\left(\mathrm{63}{x}+\mathrm{1}\right)=\mathrm{160} \\ $$$$\mathrm{63}{x}+\mathrm{5}={u} \\…

Question-210843

Question Number 210843 by peter frank last updated on 20/Aug/24 Answered by Spillover last updated on 20/Aug/24 $${let}\:{u},{v}\:{be}\:{a}\:{two}\:{unit}\:{vectors}\:{with}\:{direction} \\ $$$${cosine}\:\left({l}_{\mathrm{1}\:} ,{m}_{\mathrm{1}} ,{n}_{\mathrm{1}} \right)\:{and}\:\left({l}_{\mathrm{2}\:} ,{m}_{\mathrm{2}} ,{n}_{\mathrm{2}}…

Question-210842

Question Number 210842 by peter frank last updated on 20/Aug/24 Answered by mm1342 last updated on 20/Aug/24 $${AB}:{y}−\mathrm{2}{x}=\mathrm{1} \\ $$$${AC}:{y}−{x}=\mathrm{0} \\ $$$${BC}:−\mathrm{3}{y}+{x}=−\mathrm{4} \\ $$$$\Rightarrow{A}\left(−\mathrm{1},−\mathrm{1}\right)\:\&\:{B}\left(\frac{\mathrm{1}}{\mathrm{5}},\frac{\mathrm{7}}{\mathrm{5}}\right)\:\&\:{C}\left(\mathrm{2},\mathrm{2}\right)\Rightarrow{D}\left(\frac{\mathrm{11}}{\mathrm{10}},\frac{\mathrm{17}}{\mathrm{10}}\right) \\…