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Category: Algebra

If-0-1-then-prove-that-1-1-1-lt-1-

Question Number 176453 by mnjuly1970 last updated on 19/Sep/22 $$ \\ $$$$\:\:\:{If}\:,\:\:\alpha\:,\:\beta\:,\:\gamma\:\in\:\left(\:\mathrm{0}\:\:,\:\:\mathrm{1}\:\right)\:\:,\:\:{then}\: \\ $$$$\:\:\:\: \\ $$$$\:\:\:\:{prove}\:\:{that}\::\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\sqrt{\left(\mathrm{1}\overset{} {−}\alpha\:\right).\left(\mathrm{1}\overset{} {−}\beta\:\right).\:\left(\mathrm{1}\overset{} {−}\gamma\:\right)}\:+\sqrt{\overset{} {\alpha}.\overset{} {\beta}.\overset{}…

1-4x-4-x-2-3x-2-find-the-solution-set-2-1-cos-2-sin-2-1-cos-2-sin-2-

Question Number 110897 by bemath last updated on 31/Aug/20 $$\left(\mathrm{1}\right)\mathrm{4x}−\mathrm{4}\:\leqslant\:\mid\mathrm{x}^{\mathrm{2}} −\mathrm{3x}+\mathrm{2}\:\mid\: \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{solution}\:\mathrm{set}\: \\ $$$$\left(\mathrm{2}\right)\:\frac{\mathrm{1}+\mathrm{cos}\:\left(\frac{\alpha}{\mathrm{2}}\right)−\mathrm{sin}\:\left(\frac{\alpha}{\mathrm{2}}\right)}{\mathrm{1}−\mathrm{cos}\:\left(\frac{\alpha}{\mathrm{2}}\right)−\mathrm{sin}\:\left(\frac{\alpha}{\mathrm{2}}\right)}=? \\ $$ Answered by john santu last updated on 31/Aug/20…

Find-the-sum-of-n-terms-1-1-3-1-3-5-1-2n-1-2n-1-

Question Number 45364 by Tawa1 last updated on 12/Oct/18 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{n}\:\mathrm{terms}:\:\:\frac{\mathrm{1}}{\mathrm{1}.\mathrm{3}}\:+\:\frac{\mathrm{1}}{\mathrm{3}.\mathrm{5}}\:+\:…\:+\:\frac{\mathrm{1}}{\left(\mathrm{2n}\:−\:\mathrm{1}\right)\left(\mathrm{2n}\:+\:\mathrm{1}\right)}\:\:=\:? \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 12/Oct/18 $${T}_{{n}} =\frac{\mathrm{1}}{\left(\mathrm{2}{n}+\mathrm{1}\right)\left(\mathrm{2}{n}−\mathrm{1}\right)} \\ $$$${T}_{{n}} =\frac{\mathrm{1}}{\mathrm{2}}\left[\frac{\left(\mathrm{2}{n}+\mathrm{1}\right)−\left(\mathrm{2}{n}−\mathrm{1}\right)}{\left(\mathrm{2}{n}+\mathrm{1}\right)\left(\mathrm{2}{n}−\mathrm{1}\right)}\right] \\…

x-1-1-1-2-1-3-1-4-

Question Number 45356 by MrW3 last updated on 12/Oct/18 $${x}=\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}+\frac{\mathrm{1}}{\mathrm{3}+\frac{\mathrm{1}}{\mathrm{4}+…}}}}=? \\ $$ Answered by MJS last updated on 12/Oct/18 $$\mathrm{non}−\mathrm{periodic}\:\mathrm{infinite}\:\mathrm{continued}\:\mathrm{fractions} \\ $$$$\mathrm{are}\:\mathrm{transcendental}\:\mathrm{numbers},\:\mathrm{at}\:\mathrm{least}\:\mathrm{I}\:\mathrm{seem} \\ $$$$\mathrm{to}\:\mathrm{remember}\:\mathrm{this}.\:\mathrm{we}\:\mathrm{can}\:\mathrm{only}\:\mathrm{approximate} \\…

Question-110887

Question Number 110887 by I want to learn more last updated on 31/Aug/20 Commented by Aina Samuel Temidayo last updated on 31/Aug/20 $$\exists\:\mathrm{100a}+\mathrm{10b}+\mathrm{c}\:\in\:\left(\mathrm{100},\mathrm{999}\right)\:\mathrm{s}.\mathrm{t}. \\ $$$$\mathrm{100a}+\mathrm{10b}+\mathrm{c}\:=\:\mathrm{a}^{\mathrm{3}}…