Category: Arithmetic

Question Number 103700 by anonymous last updated on 16/Jul/20 Commented by anonymous last updated on 16/Jul/20 $${plzz}\:{help}\:{me} \\ $$ Answered by bobhans last updated on…

Question Number 103673 by bobhans last updated on 16/Jul/20 $$\underset{{k}=\mathrm{1}} {\overset{\mathrm{4095}} {\sum}}\frac{\mathrm{1}}{\left(\sqrt{{k}}+\sqrt{{k}+\mathrm{1}}\right)\left(\sqrt[{\mathrm{4}}]{{k}}+\sqrt[{\mathrm{4}}]{{k}+\mathrm{1}}\right)}\:? \\ $$ Answered by bramlex last updated on 16/Jul/20 $$\frac{\mathrm{1}}{\left(\sqrt{\mathrm{k}}+\sqrt{\mathrm{k}+\mathrm{1}}\right)\left(\sqrt[{\mathrm{4}}]{\mathrm{k}}+\sqrt[{\mathrm{4}}]{\mathrm{k}+\mathrm{1}}\right)}\:×\frac{\sqrt[{\mathrm{4}}]{\mathrm{k}+\mathrm{1}}−\sqrt[{\mathrm{4}}]{\mathrm{k}}}{\:\sqrt[{\mathrm{4}}]{\mathrm{k}+\mathrm{1}}−\sqrt[{\mathrm{4}}]{\mathrm{k}}}= \\ $$$$\frac{\sqrt[{\mathrm{4}}]{\mathrm{k}+\mathrm{1}}−\sqrt[{\mathrm{4}}]{\mathrm{k}}}{\left(\sqrt{\mathrm{k}+\mathrm{1}}+\sqrt{\mathrm{k}}\right)\left(\sqrt{\mathrm{k}+\mathrm{1}}−\sqrt{\mathrm{k}}\right)}\:=\:\sqrt[{\mathrm{4}}]{\mathrm{k}+\mathrm{1}}−\sqrt[{\mathrm{4}}]{\mathrm{k}} \\…

Question Number 103648 by Lordose last updated on 16/Jul/20 $$\boldsymbol{\mathrm{Evaluate}}\:\frac{\mathrm{1}}{\mathrm{1}\centerdot\mathrm{2}\centerdot\mathrm{3}}+\frac{\mathrm{3}}{\mathrm{2}\centerdot\mathrm{3}\centerdot\mathrm{4}}+\frac{\mathrm{5}}{\mathrm{3}\centerdot\mathrm{4}\centerdot\mathrm{5}}+…+\frac{\mathrm{2}\boldsymbol{\mathrm{n}}−\mathrm{1}}{\boldsymbol{\mathrm{n}}\left(\boldsymbol{\mathrm{n}}+\mathrm{1}\right)\left(\boldsymbol{\mathrm{n}}+\mathrm{2}\right.} \\ $$ Answered by Dwaipayan Shikari last updated on 16/Jul/20 $${T}_{{n}} =\frac{\mathrm{2}{n}−\mathrm{1}}{{n}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)} \\ $$$$\Sigma{T}_{{n}} =\Sigma\frac{{n}+{n}+\mathrm{2}−\mathrm{3}}{{n}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)}=\Sigma\frac{\mathrm{1}}{{n}\left({n}+\mathrm{1}\right)}+\Sigma\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)}−\frac{\mathrm{3}}{\mathrm{2}}\Sigma\frac{\mathrm{2}+{n}−{n}}{{n}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)}…

Question Number 38099 by Cheyboy last updated on 21/Jun/18 $${x}^{{x}} =\mathrm{0}.\mathrm{25} \\ $$$${find}\:{x} \\ $$ Commented by MrW3 last updated on 22/Jun/18 $${no}\:{real}\:{solution},\: \\ $$$${since}\:{x}^{{x}}…

Question Number 103620 by bobhans last updated on 16/Jul/20 $${If}\:{a}^{\mathrm{2}} −{bc},\:{b}^{\mathrm{2}} −{ac},\:{c}^{\mathrm{2}} −{ab}\:{is}\:{AP}\:{where}\:{a}+{c} \\ $$$$=\:\mathrm{12},\:{find}\:{the}\:{value}\:{of}\:{a}+{b}+{c}\: \\ $$ Answered by bemath last updated on 16/Jul/20 $$\Rightarrow{AP}\::\:\mathrm{2}\left({b}^{\mathrm{2}}…