Question Number 79625 by M±th+et£s last updated on 26/Jan/20 $${if}\:{n}>\mathrm{1}\:{prove}\:{that} \\ $$$$\mathrm{2}{ln}\left({n}\right)−{ln}\left({n}+\mathrm{1}\right)−{ln}\left({n}−\mathrm{1}\right)=\frac{\mathrm{1}}{{n}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{2}{n}^{\mathrm{4}} }+\frac{\mathrm{1}}{\mathrm{3}{n}^{\mathrm{6}} }+…= \\ $$ Answered by mind is power last updated on…
Question Number 14047 by FilupS last updated on 27/May/17 $${S}=\mathrm{1}+{i}−\mathrm{1}−{i}+\mathrm{1}+… \\ $$$$\frac{\mathrm{1}}{{i}}=−{i} \\ $$$${S}={i}\left(−{i}+\mathrm{1}+{i}−\mathrm{1}−{i}+\mathrm{1}+…\right) \\ $$$${S}={i}\left(−{i}+{S}\right) \\ $$$${S}=\mathrm{1}+{iS} \\ $$$${S}\left(\mathrm{1}−{i}\right)=\mathrm{1} \\ $$$$\therefore\:{S}=\frac{\mathrm{1}}{\mathrm{1}−{i}} \\ $$$$\: \\…
Question Number 79497 by john santu last updated on 25/Jan/20 $$\frac{\left(\mathrm{4}{x}−\mid{x}−\mathrm{6}\mid\right)\left(\mathrm{log}_{\frac{\mathrm{1}}{\mathrm{3}}} \left({x}+\mathrm{4}\right)+\mathrm{1}\right)}{\mathrm{2}^{{x}^{\mathrm{2}} −\mathrm{2}^{\mid{x}\mid} } }\geqslant\mathrm{0} \\ $$ Answered by john santu last updated on 25/Jan/20…
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Question Number 13909 by tawa tawa last updated on 25/May/17 $$\mathrm{Solve}\:\mathrm{simutaneously}. \\ $$$$\mathrm{x}\sqrt{\mathrm{x}}\:+\:\mathrm{y}\sqrt{\mathrm{y}}\:=\:\mathrm{183} \\ $$$$\mathrm{x}\sqrt{\mathrm{y}}\:+\:\mathrm{y}\sqrt{\mathrm{x}}\:=\:\mathrm{185} \\ $$ Commented by RasheedSindhi last updated on 25/May/17 $$\left(\mathrm{x}^{\mathrm{1}/\mathrm{2}}…
Question Number 144980 by henderson last updated on 01/Jul/21 $$\underline{\boldsymbol{\mathrm{exercise}}} \\ $$$$\boldsymbol{\mathrm{Let}}\:\boldsymbol{{a}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{{b}}\:\boldsymbol{\mathrm{be}}\:\boldsymbol{\mathrm{natural}}\:\boldsymbol{\mathrm{integers}}\:\boldsymbol{\mathrm{such}}\:\boldsymbol{\mathrm{that}}\:\mathrm{0}<\boldsymbol{{a}}<\boldsymbol{{b}}. \\ $$$$\mathrm{1}.\:\boldsymbol{\mathrm{Show}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{if}}\:\boldsymbol{{a}}\:\boldsymbol{\mathrm{divides}}\:\boldsymbol{{b}},\:\boldsymbol{\mathrm{then}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{any}}\:\boldsymbol{\mathrm{naturel}}\: \\ $$$$\boldsymbol{\mathrm{number}}\:\boldsymbol{{n}},\:\boldsymbol{{n}}^{\boldsymbol{{a}}} −\mathrm{1}\:\boldsymbol{\mathrm{divides}}\:\boldsymbol{{n}}^{\boldsymbol{{b}}} −\mathrm{1}. \\ $$$$ \\ $$$$\mathrm{2}.\:\boldsymbol{\mathrm{For}}\:\boldsymbol{\mathrm{any}}\:\boldsymbol{\mathrm{non}}−\boldsymbol{\mathrm{zero}}\:\boldsymbol{\mathrm{naturel}}\:\boldsymbol{\mathrm{number}}\:\boldsymbol{{n}},\:\boldsymbol{\mathrm{prove}}\: \\ $$$$\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{remainder}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{euclidean}}\:\boldsymbol{\mathrm{division}}\:\boldsymbol{\mathrm{of}}\: \\…
Question Number 144928 by ArielVyny last updated on 30/Jun/21 $$\underset{{n}\geqslant\mathrm{0}} {\sum}\frac{\left(−\mathrm{1}\right)^{{n}} }{\left({n}+{z}\right){n}!} \\ $$ Answered by Dwaipayan Shikari last updated on 01/Jul/21 $$\int_{\mathrm{0}} ^{\mathrm{1}} \underset{{n}\geqslant\mathrm{0}}…
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Question Number 78979 by jagoll last updated on 22/Jan/20 $$\mathrm{the}\:\mathrm{first}\:\mathrm{n}\:\mathrm{even} \\ $$$$\mathrm{numbers}\:\mathrm{are}\:\mathrm{known}.\:\mathrm{if}\:\mathrm{one}\:\mathrm{number} \\ $$$$\mathrm{is}\:\mathrm{deleted}\:\mathrm{then}\:\mathrm{the}\:\mathrm{average} \\ $$$$\mathrm{of}\:\mathrm{remaining}\:\mathrm{numbers}\:\mathrm{is}\:\frac{\mathrm{2582}}{\mathrm{50}}. \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{deleted}\:\mathrm{number}. \\ $$ Commented by mr W last…
Question Number 144482 by ArielVyny last updated on 25/Jun/21 $${show}\:{that}\:\forall{n}\in\mathbb{Z}\: \\ $$$${E}\left(\frac{{n}−\mathrm{1}}{\mathrm{2}}\right)+{E}\left(\frac{{n}+\mathrm{2}}{\mathrm{4}}\right)+{E}\left(\frac{{n}+\mathrm{4}}{\mathrm{4}}\right)={n} \\ $$ Answered by Olaf_Thorendsen last updated on 25/Jun/21 $${x}\:=\:\mathrm{E}\left(\frac{{n}−\mathrm{1}}{\mathrm{2}}\right)+\mathrm{E}\left(\frac{{n}+\mathrm{2}}{\mathrm{4}}\right)+\mathrm{E}\left(\frac{{n}+\mathrm{4}}{\mathrm{4}}\right) \\ $$$$\mathrm{1st}\:\mathrm{case}\::\:{n}\:=\:\mathrm{4}{p} \\…