Question Number 25482 by math solver last updated on 11/Dec/17 Commented by prakash jain last updated on 11/Dec/17 $${x}=\frac{\sqrt{\mathrm{3}}\pm\sqrt{\mathrm{3}−\mathrm{4}}}{\mathrm{2}}=\frac{\sqrt{\mathrm{3}}\pm{i}}{\mathrm{2}} \\ $$$${x}=\frac{\sqrt{\mathrm{3}}+{i}}{\mathrm{2}},\:\omega=\frac{−\mathrm{1}+{i}\sqrt{\mathrm{3}}}{\mathrm{2}}\Rightarrow{x}=−{i}\omega \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\mathrm{24}} {\sum}}\left(−{iw}\right)^{{n}}…
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Question Number 25462 by Tinkutara last updated on 10/Dec/17 $$\mathrm{Let}\:{S}_{{n}} ,\:{n}\:=\:\mathrm{1},\:\mathrm{2},\:\mathrm{3}…\:\mathrm{be}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of} \\ $$$$\mathrm{infinite}\:\mathrm{geometric}\:\mathrm{series}\:\mathrm{whose}\:\mathrm{first} \\ $$$$\mathrm{term}\:\mathrm{is}\:{n}\:\mathrm{and}\:\mathrm{the}\:\mathrm{common}\:\mathrm{ratio}\:\mathrm{is} \\ $$$$\frac{\mathrm{1}}{{n}\:+\:\mathrm{1}}.\:\mathrm{Then} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{{S}_{\mathrm{1}} {S}_{{n}} \:+\:{S}_{\mathrm{2}} {S}_{{n}−\mathrm{1}} \:+\:{S}_{\mathrm{3}} {S}_{{n}−\mathrm{2}}…
Question Number 156532 by MathSh last updated on 12/Oct/21 Answered by Rasheed.Sindhi last updated on 13/Oct/21 $$\:\:\:\:\:\mathrm{When}\:\mathrm{f}\left(\mathrm{x}\right),\mathrm{g}\left(\mathrm{x}\right)\:\mathrm{are}\:\boldsymbol{\mathrm{constant}}\:\mathrm{or} \\ $$$$\:\:\:\:\:\:\boldsymbol{\mathrm{linear}}. \\ $$$$\begin{cases}{\mathrm{f}\left(\:\mathrm{x}+\mathrm{g}\left(\mathrm{x}\right)\:\right)=\mathrm{g}\left(\:\mathrm{x}+\mathrm{f}\left(\mathrm{x}\right)\:\right)}\\{\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\:\mathrm{f}\left(\mathrm{x}\right)\:\right)=\mathrm{g}\left(\mathrm{x}\right)+\mathrm{f}\left(\:\mathrm{g}\left(\mathrm{x}\right)\:\right)}\end{cases}\:\: \\ $$$$\mathrm{Assuming}\:\mathrm{that}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{and}\:\mathrm{g}\left(\mathrm{x}\right)\:\mathrm{are}\:\mathrm{both} \\ $$$$\mathrm{linear}:\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{ax}+\mathrm{b}\:\&\:\mathrm{g}\left(\mathrm{x}\right)=\mathrm{cx}+\mathrm{d}…
Question Number 156535 by MathSh last updated on 12/Oct/21 Answered by Rasheed.Sindhi last updated on 12/Oct/21 $$\mathrm{Let}\:\sqrt{\mathrm{x}}=\mathrm{a},\sqrt{\mathrm{y}}=\mathrm{b},\sqrt{\mathrm{z}}=\mathrm{c} \\ $$$$\mathrm{a}^{\mathrm{4}} \mathrm{bc}+\mathrm{b}^{\mathrm{4}} \mathrm{ac}+\mathrm{c}^{\mathrm{4}} \mathrm{ab}−\mathrm{3a}^{\mathrm{2}} \mathrm{b}^{\mathrm{2}} \mathrm{c}^{\mathrm{2}} =\mathrm{0}…
Question Number 25457 by Tinkutara last updated on 10/Dec/17 Answered by prakash jain last updated on 10/Dec/17 $${n}^{\mathrm{2}} +{n}+\mathrm{1}={an}\left({n}−\mathrm{1}\right)+{bn}+{c} \\ $$$${n}\left({n}−\mathrm{1}\right)+\mathrm{2}{n}+\mathrm{1} \\ $$$$\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{i}}…
Question Number 156513 by cortano last updated on 12/Oct/21 $$\mathrm{x}^{\mathrm{4}} +\mathrm{4x}^{\mathrm{3}} −\mathrm{26x}^{\mathrm{2}} −\mathrm{20x}+\mathrm{25}=\mathrm{0} \\ $$$$\mathrm{x}=?\:\left(\mathrm{exact}\:\mathrm{form}\right) \\ $$ Commented by john_santu last updated on 12/Oct/21 $$\Rightarrow{x}^{\mathrm{2}}…
Question Number 25444 by math solver last updated on 10/Dec/17 Commented by Rasheed.Sindhi last updated on 16/Dec/17 $$\bar {\mathrm{z}}+\mathrm{1}=\mathrm{iz}^{\mathrm{2}} +\mid\mathrm{z}\mid^{\mathrm{2}} \\ $$$$\mathrm{z}=\mathrm{x}+\mathrm{iy}\Rightarrow\overline {\mathrm{x}+\mathrm{iy}}+\mathrm{1}=\mathrm{i}\left(\mathrm{x}+\mathrm{iy}\right)^{\mathrm{2}} +\mid\mathrm{x}+\mathrm{iy}\mid \\…