Question Number 175218 by mnjuly1970 last updated on 23/Aug/22 $$ \\ $$$$\:\:\lfloor\:\:\left(\:\mathrm{1}\:+\:\sqrt{\mathrm{5}}\:\right)^{\:\mathrm{8}} \:\rfloor\:=\:? \\ $$$$\:\:\: \\ $$ Answered by behi834171 last updated on 23/Aug/22 $$\varphi^{\mathrm{2}}…
Question Number 175180 by Rasheed.Sindhi last updated on 22/Aug/22 $${a}_{{n}} \:{is}\:{an}\:{AP}\:{and}\:{S}_{{n}} \:{is}\:{sum}\:{of}\:{n}\:{terms} \\ $$$${of}\:{this}\:{AP}. \\ $$$${Given}\:{that}\:{S}_{\mathrm{11}} −{S}_{\mathrm{7}} =\mathrm{72},\:{determine} \\ $$$${a}_{\mathrm{6}} +{a}_{\mathrm{13}} . \\ $$ Answered…
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Question Number 175175 by Shrinava last updated on 21/Aug/22 Answered by TheHoneyCat last updated on 23/Aug/22 $$\mathrm{Let}\:\mathrm{me}\:\mathrm{proove}\:\mathrm{a}\:\mathrm{slightly}\:\mathrm{more}\:\mathrm{general}\:\mathrm{statement}: \\ $$$$\mathrm{let}\:{x}=\frac{{a}}{\mathrm{2}}\:\leqslant\:{y}=\frac{{c}}{\mathrm{2}}\: \\ $$$$\mathrm{let}\:{t}\in\left[\mathrm{0},\mathrm{1}\right]\::\:{t}.{x}+\left(\mathrm{1}−{t}\right).{y}={b} \\ $$$$\mathrm{and}\:\mathrm{let}\:\mathrm{us}\:\mathrm{proove}\:\mathrm{that} \\ $$$$\mathrm{e}^{{x}}…
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Question Number 44095 by peter frank last updated on 21/Sep/18 Answered by tanmay.chaudhury50@gmail.com last updated on 22/Sep/18 $${ax}^{\mathrm{3}} +{bx}^{\mathrm{2}} +{cx}+{d}=\mathrm{0} \\ $$$$\alpha+\beta+\gamma=\frac{−{b}}{{a}} \\ $$$$\alpha\beta+\beta\gamma\alpha\gamma=\frac{{c}}{{a}} \\…
Question Number 109626 by qwerty111 last updated on 24/Aug/20 Commented by kaivan.ahmadi last updated on 24/Aug/20 $${log}_{\mathrm{20}} \mathrm{2}×\mathrm{5}^{\mathrm{3}} ={log}_{\mathrm{20}} \mathrm{2}+\mathrm{3}{log}_{\mathrm{20}} \mathrm{5}=\frac{\mathrm{1}}{{log}_{\mathrm{2}} \mathrm{20}}+\frac{\mathrm{3}}{{log}_{\mathrm{5}} \mathrm{20}}= \\ $$$$\frac{\mathrm{1}}{\mathrm{2}+{log}_{\mathrm{2}}…
Question Number 109625 by qwerty111 last updated on 24/Aug/20 $$\frac{\mathrm{sin}\:\mathrm{2}\boldsymbol{\alpha}+\mathrm{2sin}\:\boldsymbol{\alpha}\centerdot\mathrm{cos}\:\mathrm{2}\boldsymbol{\alpha}}{\mathrm{1}+\mathrm{cos}\:\boldsymbol{\alpha}+\mathrm{cos2}\:\boldsymbol{\alpha}+\mathrm{cos3}\:\boldsymbol{\alpha}} \\ $$ Commented by nimnim last updated on 24/Aug/20 $${tan}\alpha \\ $$ Commented by kaivan.ahmadi…
Question Number 175154 by Shrinava last updated on 21/Aug/22 Commented by mr W last updated on 21/Aug/22 $${i}\:{don}'{t}\:{think}\:{there}\:{are}\:{unique}\:{solutions}. \\ $$ Answered by MJS_new last updated…
Question Number 175148 by Shrinava last updated on 20/Aug/22 Commented by mr W last updated on 21/Aug/22 $${wrong}! \\ $$$${for}\:{a}=−\mathrm{1},\:{b}=\mathrm{1} \\ $$$$\mathrm{2}^{{b}} −\mathrm{2}^{{a}} =\mathrm{2}−\frac{\mathrm{1}}{\mathrm{2}}=\frac{\mathrm{3}}{\mathrm{2}}=\mathrm{1}.\mathrm{5}>\frac{\mathrm{9}\:\mathrm{log}\:\mathrm{2}}{\mathrm{4}}\approx\mathrm{0}.\mathrm{677} \\…