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Category: Algebra

Question-119107

Question Number 119107 by I want to learn more last updated on 22/Oct/20 Answered by 1549442205PVT last updated on 22/Oct/20 $$\mathrm{S}=\mathrm{1}+\mathrm{3}−\mathrm{5}+\mathrm{7}+\mathrm{9}−\mathrm{11}+\mathrm{13}+\mathrm{15}−\mathrm{17}+\mathrm{19}+\mathrm{21}−\mathrm{23}+… \\ $$$$\mathrm{S}=\left(\mathrm{1}+\mathrm{3}−\mathrm{5}\right)+\left(\mathrm{7}+\mathrm{9}−\mathrm{11}\right)+\left(\mathrm{13}+\mathrm{15}−\mathrm{17}\right)+\left(\mathrm{19}+\mathrm{21}−\mathrm{23}\right)+ \\ $$$$\left(\mathrm{25}+\mathrm{27}−\mathrm{29}\right)+…+\left[\mathrm{6n}−\mathrm{5}+\mathrm{6n}−\mathrm{3}−\left(\mathrm{6n}−\mathrm{5}\right)\right]…

Solve-for-real-numbers-sinx-1-sin-2-x-1-cosy-1-cos-2-y-

Question Number 184622 by Shrinava last updated on 09/Jan/23 $$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\mathrm{sinx}\:\sqrt{\mathrm{1}\:−\:\mathrm{sin}^{\mathrm{2}} \mathrm{x}}\:=\:\mathrm{1}\:+\:\mathrm{cosy}\:\sqrt{\mathrm{1}\:−\:\mathrm{cos}^{\mathrm{2}} \mathrm{y}}\: \\ $$ Answered by floor(10²Eta[1]) last updated on 09/Jan/23 $$\mathrm{sinxcosx}=\mathrm{1}+\mathrm{senycosy} \\…

Question-119059

Question Number 119059 by qqqqqqqqq last updated on 21/Oct/20 Answered by MJS_new last updated on 21/Oct/20 $$\sqrt{{x}^{\mathrm{2}} +{x}^{\mathrm{2}} \left({x}+\mathrm{1}\right)^{\mathrm{2}} +\left({x}+\mathrm{1}\right)^{\mathrm{2}} }−{x}^{\mathrm{2}} = \\ $$$$=\sqrt{{x}^{\mathrm{4}} +\mathrm{2}{x}^{\mathrm{3}}…

Question-119052

Question Number 119052 by shahria14 last updated on 21/Oct/20 Answered by 1549442205PVT last updated on 22/Oct/20 $$\mid\mathrm{x}^{\mathrm{2}} −\mathrm{1}\mid\leqslant\mathrm{3}\Leftrightarrow−\mathrm{3}\leqslant\mathrm{x}^{\mathrm{2}} −\mathrm{1}\leqslant\mathrm{3}\Leftrightarrow−\mathrm{2}\leqslant\mathrm{x}^{\mathrm{2}} \leqslant\mathrm{4} \\ $$$$\Leftrightarrow\mathrm{x}^{\mathrm{2}} \leqslant\mathrm{4}\Leftrightarrow\mid\mathrm{x}\mid\leqslant\mathrm{2}\Leftrightarrow−\mathrm{2}\leqslant\mathrm{x}\leqslant\mathrm{2} \\ $$$$…

Show-by-recurence-that-n-N-k-1-n-1-k-k-1-k-2-n-n-3-4-n-1-n-2-

Question Number 119055 by mathocean1 last updated on 21/Oct/20 $$ \\ $$$${Show}\:{by}\:{recurence}\:{that}:\:\forall\:{n}\in\mathbb{N}^{\ast} \\ $$$$\sum_{{k}=\mathrm{1}} ^{{n}} \frac{\mathrm{1}}{{k}\left({k}+\mathrm{1}\right)\left({k}+\mathrm{2}\right)}=\frac{{n}\left({n}+\mathrm{3}\right)}{\mathrm{4}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)} \\ $$ Answered by Bird last updated on 21/Oct/20…