Question Number 227139 by Spillover last updated on 02/Jan/26 Answered by MrAjder last updated on 03/Jan/26 $$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{2}{n}\left(\mathrm{2}{n}−\mathrm{1}\right)}\overset{\left[\mathrm{1}\right]} {=}\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{1}}{\mathrm{2}{n}−\mathrm{1}}−\frac{\mathrm{1}}{\mathrm{2}{n}}\right) \\ $$$$\overset{\left[\mathrm{1}\right]} {=}\underset{{n}=\mathrm{1}}…
Question Number 226943 by Spillover last updated on 20/Dec/25 Commented by fantastic2 last updated on 26/Dec/25 $${the}\:{rectangle}\:{is}\:{not}\:{unique} \\ $$ Commented by fantastic2 last updated on…
Question Number 226942 by Spillover last updated on 20/Dec/25 Answered by Spillover last updated on 24/Dec/25 Answered by Spillover last updated on 24/Dec/25 Answered by…
Question Number 226919 by Spillover last updated on 19/Dec/25 Answered by Ghisom_ last updated on 19/Dec/25 $$\underset{{n}} {\underbrace{\mathrm{6}…}}\mathrm{8}^{\mathrm{2}} =\underset{{n}} {\underbrace{\mathrm{4}…}}\mathrm{6}\underset{{n}} {\underbrace{\mathrm{2}…}}\mathrm{4} \\ $$$$\Rightarrow\:\underset{{digits}} {\sum}=\left(\mathrm{4}+\mathrm{2}\right){n}+\mathrm{6}+\mathrm{4}=\mathrm{6}{n}+\mathrm{10} \\…
Question Number 226798 by aba_math last updated on 15/Dec/25 $${let}\:{gcd}\left({m},{n}\right)=\mathrm{1}.\:{Determine}\:{gcd}\left(\mathrm{5}^{{m}} +\mathrm{7}^{{m}} ,\mathrm{5}^{{n}} +\mathrm{7}^{{n}} \right) \\ $$ Answered by Frix last updated on 15/Dec/25 $${a},\:{b},\:{c},\:…\:\in\mathbb{N} \\…
Question Number 226779 by Spillover last updated on 14/Dec/25 $${By}\:{using}\:{De}\:{Moivres}\:{theorm} \\ $$$${simplify} \\ $$$$\left({a}\right)\frac{\left(\mathrm{cos}\:\frac{\pi}{\mathrm{2}}−{i}\mathrm{sin}\:\frac{\pi}{\mathrm{2}}\right)\left(\mathrm{cos}\:\frac{\pi}{\mathrm{3}}+{i}\mathrm{sin}\:\frac{\pi}{\mathrm{3}}\right)}{\mathrm{cos}\:\frac{\pi}{\mathrm{3}}−{i}\mathrm{sin}\:\frac{\pi}{\mathrm{3}}} \\ $$$$\left({b}\right)\frac{\mathrm{cos}\:\frac{\pi}{\mathrm{8}}+{i}\mathrm{sin}\:\frac{\pi}{\mathrm{8}}}{\mathrm{cos}\:\frac{\pi}{\mathrm{6}}+{i}\mathrm{sin}\:\frac{\pi}{\mathrm{6}}} \\ $$ Answered by Frix last updated on 14/Dec/25…
Question Number 226775 by aba_math last updated on 14/Dec/25 $${Prove}\:{that}\:\left({a}−{b}\right)\left({a}−{c}\right)\left({a}−{d}\right)\left({b}−{c}\right)\left({b}−{d}\right)\left({c}−{d}\right)\:{divisible}\:{by}\:\mathrm{12} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 226721 by Spillover last updated on 11/Dec/25 Answered by breniam last updated on 12/Dec/25 $${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{2}{xyy}'=\mathrm{0} \\ $$$${t}=\mathrm{ln}\mid{x}\mid \\ $$$${x}={e}^{{t}} \\ $$$${y}\left({x}\right)={z}\left({t}\right)…
Question Number 226697 by Linton last updated on 10/Dec/25 Commented by Linton last updated on 10/Dec/25 $${The}\:{letters}\:{in}\:{TWO}\:{UV}\:{PAIRS}\: \\ $$$${have}\:{the}\:{values}\:\mathrm{0}\:\mathrm{1}\:\mathrm{2}…\:\mathrm{9}\:{in}\:{some} \\ $$$${order}\:{with}\:{each}\:{letter}\:{represent} \\ $$$${ing}\:{a}\:{different}\:{digit}. \\ $$…
Question Number 226612 by Spillover last updated on 07/Dec/25 Commented by SonGoku last updated on 07/Dec/25 $$\boldsymbol{\mathrm{Solve}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{system}}\:\begin{cases}{\boldsymbol{\mathrm{m}}\:+\:\boldsymbol{\mathrm{m}}\:=\:\mathrm{20}}\\{\boldsymbol{\mathrm{c}}\:−\:\boldsymbol{\mathrm{l}}\:=\:\mathrm{3}}\\{\boldsymbol{\mathrm{m}}\:−\:\boldsymbol{\mathrm{l}}×\boldsymbol{\mathrm{c}}\:=\:?}\end{cases}. \\ $$ Terms of Service Privacy Policy Contact:…