Category: Arithmetic

Question Number 194015 by Rupesh123 last updated on 26/Jun/23 Answered by witcher3 last updated on 26/Jun/23 $$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)=\underset{\mathrm{k}\geqslant\mathrm{1}} {\sum}\mathrm{f}_{\mathrm{k}} \mathrm{x}^{\mathrm{k}} \\ $$$${f}_{{k}+\mathrm{1}} ={f}_{{k}} +{f}_{{k}−\mathrm{1}} ;\mathrm{f}_{\mathrm{1}} =\mathrm{f}_{\mathrm{2}}…

Question Number 193853 by York12 last updated on 21/Jun/23 $$ \\ $$$$\boldsymbol{{Let}}\:\boldsymbol{{n}}\:\&\:\boldsymbol{{k}}\:\boldsymbol{{be}}\:\boldsymbol{{positive}}\:\boldsymbol{{integers}}\:\boldsymbol{{and}}\:\boldsymbol{{let}} \\ $$$$\boldsymbol{{S}}\:\boldsymbol{{be}}\:\boldsymbol{{a}}\:\boldsymbol{{set}}\:\boldsymbol{{of}}\:\boldsymbol{{n}}\:\boldsymbol{{points}}\:\boldsymbol{{in}}\:\boldsymbol{{The}}\:\boldsymbol{{plane}}\:\boldsymbol{{such}}\:\boldsymbol{{that}}\:: \\ $$$$\boldsymbol{{For}}\:\boldsymbol{{any}}\:\boldsymbol{{point}}\:\boldsymbol{{P}}\:\boldsymbol{{of}}\:\boldsymbol{{S}}\:\boldsymbol{{there}}\:\boldsymbol{{are}}\:\boldsymbol{{at}}\:\boldsymbol{{least}}\:\boldsymbol{{K}}\:\boldsymbol{{points}}\:\boldsymbol{{of}}\:\boldsymbol{{S}}\:\boldsymbol{{Equidistant}}\:\boldsymbol{{from}}\:\boldsymbol{{p}} \\ $$$$\boldsymbol{{Prove}}\:\boldsymbol{{that}}\:\boldsymbol{{k}}<\frac{\mathrm{1}}{\mathrm{2}}+\sqrt{\mathrm{2}\boldsymbol{{n}}} \\ $$ Terms of Service Privacy Policy…

Question Number 193760 by York12 last updated on 19/Jun/23 $${prove}\:{that} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}×\frac{\mathrm{3}}{\mathrm{4}}×\frac{\mathrm{5}}{\mathrm{6}}×\frac{\mathrm{7}}{\mathrm{8}}×\frac{\mathrm{9}}{\mathrm{10}}…×\frac{\mathrm{99}}{\mathrm{100}}>\frac{\mathrm{1}}{\mathrm{13}}\:\:\: \\ $$ Answered by MM42 last updated on 19/Jun/23 $${can}\:{br}\:{show}\::\:\underset{{i}=\mathrm{1}} {\overset{{n}} {\prod}}\:\frac{\mathrm{2}{i}−\mathrm{1}}{\mathrm{2}{i}}\:>\:\frac{\mathrm{11}}{\mathrm{10}\sqrt{\mathrm{4}{n}+\mathrm{1}}} \\…

Question Number 193438 by Mingma last updated on 14/Jun/23 Answered by qaz last updated on 14/Jun/23 $${log}_{\mathrm{3}} \left(\mathrm{9}{x}−\mathrm{3}\right)=\mathrm{1}+{log}_{\mathrm{3}} \left(\mathrm{3}{x}−\mathrm{1}\right)\:\:\:\:\:,{log}_{\mathrm{3}} \left({x}−\frac{\mathrm{1}}{\mathrm{3}}\right)={log}_{\mathrm{3}} \left(\mathrm{3}{x}−\mathrm{1}\right)−\mathrm{1} \\ $$$${log}_{\mathrm{3}} \left(\mathrm{3}{x}−\mathrm{1}\right)={y} \\…

Question Number 193280 by Mingma last updated on 09/Jun/23 Answered by AST last updated on 09/Jun/23 $$\mathrm{1000}\overset{\_\_\_\_} {{def}}+\overset{\_\_\_\_} {{abc}}=\mathrm{6000}\overset{\_\_\_\_} {{abc}}+\mathrm{6}\overset{\_\_\_\_} {{def}}\Rightarrow\frac{{def}}{{abc}}=\frac{\mathrm{5999}}{\mathrm{994}}=\frac{\mathrm{857}}{\mathrm{142}} \\ $$$$\Rightarrow\overset{\_\_\_\_\_\_\_\_} {{abcdef}}=\mathrm{142857} \\…