Question Number 150374 by mathdanisur last updated on 11/Aug/21 $$\underset{\boldsymbol{\mathrm{k}}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{2}^{\boldsymbol{\mathrm{k}}} \:+\:\mathrm{3}^{\boldsymbol{\mathrm{k}}} }{\mathrm{5}^{\boldsymbol{\mathrm{k}}} }\:\:=\:? \\ $$ Answered by eman_64 last updated on 11/Aug/21 $$\:\:\:=\:\sum_{\mathrm{k}=\mathrm{0}}…
Question Number 84835 by Power last updated on 16/Mar/20 Commented by abdomathmax last updated on 16/Mar/20 $${S}=\sum_{{k}=\mathrm{1}} ^{\mathrm{32}} \:\frac{{k}}{{k}+\mathrm{1}}\:=\sum_{{k}=\mathrm{1}} ^{\mathrm{32}} \:\frac{{k}+\mathrm{1}−\mathrm{1}}{{k}+\mathrm{1}} \\ $$$$=\sum_{{k}=\mathrm{1}} ^{\mathrm{32}} \left(\mathrm{1}\right)−\sum_{{k}=\mathrm{1}}…
Question Number 84820 by Power last updated on 16/Mar/20 Answered by MJS last updated on 16/Mar/20 $$\mathrm{super}\:\mathrm{easy} \\ $$$$\sqrt{\mathrm{2}{x}−\mathrm{5}}={t}\:\Leftrightarrow\:{x}=\frac{\mathrm{1}}{\mathrm{2}}{t}^{\mathrm{2}} +\frac{\mathrm{5}}{\mathrm{2}}\wedge{t}\geqslant\mathrm{0} \\ $$$$\sqrt{\frac{\mathrm{1}}{\mathrm{2}}{t}^{\mathrm{2}} +\frac{\mathrm{5}}{\mathrm{2}}−\mathrm{2}+{t}}+\sqrt{\frac{\mathrm{1}}{\mathrm{2}}{t}^{\mathrm{2}} +\frac{\mathrm{5}}{\mathrm{2}}+\mathrm{2}+\mathrm{3}{t}}=\mathrm{7}\sqrt{\mathrm{2}} \\…
Question Number 150350 by mathdanisur last updated on 11/Aug/21 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 150346 by mathdanisur last updated on 11/Aug/21 $$\mathrm{If}\:\:\boldsymbol{\mathrm{x}};\boldsymbol{\mathrm{y}}\:\mathrm{and}\:\boldsymbol{\mathrm{z}}\:\mathrm{are}\:\mathrm{positive}\:\mathrm{integers},\:\mathrm{then} \\ $$$$\mathrm{determine}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{positive}\:\mathrm{integer} \\ $$$$\boldsymbol{\mathrm{N}}=\mathrm{x}+\mathrm{y}+\mathrm{z}+\mathrm{xy}+\mathrm{yz}+\mathrm{zx},\:\mathrm{which}\:\mathrm{is} \\ $$$$\mathrm{bigger}\:\mathrm{than}\:\mathrm{2022}. \\ $$ Commented by Rasheed.Sindhi last updated on 12/Aug/21…
Question Number 150328 by mathdanisur last updated on 11/Aug/21 $$\mathrm{Find}\:\mathrm{all}\:\mathrm{the}\:\mathrm{solutions}\:\mathrm{to}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\boldsymbol{\mathrm{x}}^{\mathrm{3}} \:−\:\boldsymbol{\mathrm{y}}^{\mathrm{3}} \:=\:\boldsymbol{\mathrm{xy}}\:+\:\mathrm{61} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 150332 by eman_64 last updated on 11/Aug/21 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 84792 by M±th+et£s last updated on 16/Mar/20 $${a},{b},{c}\geqslant\mathrm{0} \\ $$$${a}+{b}+{c}=\mathrm{3} \\ $$$${show}\:{that} \\ $$$$\sqrt[{\mathrm{3}}]{{a}}\:+\sqrt[{\mathrm{3}}]{{b}}\:+\sqrt[{\mathrm{3}}]{{c}}\geqslant{ab}+{bc}+{ca} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 150331 by ajfour last updated on 11/Aug/21 $${ultimately}\:\:{Q}.\mathrm{149894}\:\:{boils}\: \\ $$$${down}\:{to}\:{finding}\:{s}_{{max}} ,\:{s}_{{min}} \:\forall \\ $$$$\:\:\:\:\left\{{h}−{s}\mathrm{cos}\:\left(\theta−\frac{\pi}{\mathrm{6}}\right)\right\}^{\mathrm{2}} \\ $$$$+\left\{{k}−{s}\mathrm{sin}\:\left(\theta+\frac{\pi}{\mathrm{6}}\right)\right\}^{\mathrm{2}} ={r}^{\mathrm{2}} \\ $$$$ \\ $$ Terms of…
Question Number 150327 by mathdanisur last updated on 11/Aug/21 $$\left.\mathrm{1}\right)\:\mathrm{33}\boldsymbol{\mathrm{x}}\:\:\equiv\:\:\mathrm{48}\:\left(\mathrm{mod}\:\mathrm{654}\right) \\ $$$$\left.\mathrm{2}\right)\:\mathrm{5}^{\mathrm{1000}\:\mathrm{000}} \:\:\equiv\:\:\boldsymbol{\mathrm{x}}\:\left(\mathrm{mod}\:\mathrm{41}\right) \\ $$$$\mathrm{Find}\:\:\boldsymbol{\mathrm{x}}=? \\ $$ Answered by Rasheed.Sindhi last updated on 11/Aug/21 $$\left.\mathrm{1}\right)\:\mathrm{33}\boldsymbol{\mathrm{x}}\:\:\equiv\:\:\mathrm{48}\:\left(\mathrm{mod}\:\mathrm{654}\right)…