Question Number 206365 by hardmath last updated on 12/Apr/24 $$\mathrm{Number}\:\mathrm{series}: \\ $$$$\mathrm{a}_{\mathrm{3}} \:=\:\mathrm{2a}\:+\:\mathrm{b}\:−\:\mathrm{6} \\ $$$$\mathrm{a}_{\mathrm{9}} \:=\:\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{5} \\ $$$$\mathrm{a}_{\mathrm{15}} \:=\:\mathrm{3a}\:+\:\mathrm{b}\:−\:\mathrm{7} \\ $$$$\mathrm{Find}:\:\:\:\mathrm{a}\:=\:? \\ $$ Commented by…
Question Number 206332 by MATHEMATICSAM last updated on 12/Apr/24 $$\mathrm{If}\:\mathrm{log}\left({a}\:+\:{b}\:+\:{c}\right)\:=\:\mathrm{log}{a}\:+\:\mathrm{log}{b}\:+\:\mathrm{log}{c}\: \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\mathrm{log}\left(\frac{\mathrm{2}{a}}{\mathrm{1}\:−\:{a}^{\mathrm{2}} }\:+\:\frac{\mathrm{2}{b}}{\mathrm{1}\:−\:{b}^{\mathrm{2}} }\:+\:\frac{\mathrm{2}{c}}{\mathrm{1}\:−\:{c}^{\mathrm{2}} }\right)\:=\: \\ $$$$\mathrm{log}\left(\frac{\mathrm{2}{a}}{\mathrm{1}\:−\:{a}^{\mathrm{2}} }\right)\:+\:\mathrm{log}\left(\frac{\mathrm{2}{b}}{\mathrm{1}\:−\:{b}^{\mathrm{2}} }\right)\:+\:\mathrm{log}\left(\frac{\mathrm{2}{c}}{\mathrm{1}\:−\:{c}^{\mathrm{2}} }\right). \\ $$ Answered…
Question Number 206328 by mnjuly1970 last updated on 12/Apr/24 $$ \\ $$$$\:\:\:\:\:\:\:{f}\left({x}\right)=\:\lfloor\:\frac{\:\mathrm{1}+\:{x}\:+{x}^{\mathrm{2}} }{{x}+\:{x}^{\:\mathrm{2}} }\:\rfloor\:{is}\:{given}. \\ $$$$\:\:\:\:\:\:\:\Rightarrow\begin{cases}{\:\:{D}_{{f}} \:=\:?\:\left({domain}\:\right)}\\{\:\:\:{R}_{\:{f}} \:=\:?\left(\:{range}\right)}\end{cases} \\ $$$$ \\ $$ Answered by TheHoneyCat…
Question Number 206367 by hardmath last updated on 12/Apr/24 $$\mathrm{Find}: \\ $$$$\frac{\mathrm{1}}{\mathrm{6}}\:\:+\:\:\frac{\mathrm{1}}{\mathrm{24}}\:\:+\:\:\frac{\mathrm{1}}{\mathrm{60}}\:\:+\:…\:+\:…\:\:\frac{\mathrm{1}}{\mathrm{720}}\:=\:? \\ $$ Commented by TheHoneyCat last updated on 12/Apr/24 Can you please explain what the "..." are? Commented by TonyCWX08…
Question Number 206363 by hardmath last updated on 12/Apr/24 $$\mathrm{If}\:\:\:\mathrm{0}<\mathrm{a}<\mathrm{1} \\ $$$$\mathrm{Compare}: \\ $$$$\frac{\mathrm{1}}{\mathrm{a}−\mathrm{1}}\:\:,\:\:\frac{\mathrm{a}}{\mathrm{a}−\mathrm{1}}\:\:,\:\:\frac{\mathrm{1}}{\mathrm{1}−\mathrm{a}}\:\:,\:\:\frac{\mathrm{a}}{\mathrm{1}−\mathrm{a}}\:\:,\:\:\frac{\mathrm{a}}{\mathrm{2a}} \\ $$ Commented by TheHoneyCat last updated on 12/Apr/24 Commented by…
Question Number 206357 by hardmath last updated on 12/Apr/24 $$\mathrm{Find}: \\ $$$$\mathrm{1}\:+\:\mathrm{cos444}°\:−\:\mathrm{cos84}°\:+\:\mathrm{cot45}°\:=\:? \\ $$ Answered by MM42 last updated on 12/Apr/24 $$\mathrm{2}\:\checkmark \\ $$ Commented…
Question Number 206322 by MATHEMATICSAM last updated on 12/Apr/24 $$\mathrm{If}\:{abc}\:=\:\mathrm{1}\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\:\frac{\mathrm{1}}{\mathrm{1}\:+\:{a}\:+\:{b}^{−\mathrm{1}} }\:+\:\frac{\mathrm{1}}{\mathrm{1}\:+\:{b}\:+\:{c}^{−\mathrm{1}} }\:+\:\frac{\mathrm{1}}{\mathrm{1}\:+\:{c}\:+\:{a}^{−\mathrm{1}} }\:=\:\mathrm{1} \\ $$ Commented by Rasheed.Sindhi last updated on 12/Apr/24 $${For}\:{similar}\:{question}\:{see}…
Question Number 206355 by mr W last updated on 12/Apr/24 $${if}\:{the}\:{sum}\:{of}\:{three}\:{positive}\:{real}\: \\ $$$${numbers}\:{is}\:{equal}\:{to}\:{their}\:{product}, \\ $$$${prove}\:{that}\:{at}\:{least}\:{one}\:{of}\:{the}\: \\ $$$${numbers}\:{is}\:{larger}\:{than}\:\mathrm{1}.\mathrm{7}. \\ $$ Answered by A5T last updated on…
Question Number 206339 by mnjuly1970 last updated on 12/Apr/24 $$ \\ $$$$\:\:\:\:\:{E}\:\subseteq\:{Y}\:\subseteq\:\left(\:{X}\:,\:{d}\:\right)\mid_{{metric}\:{space}} \\ $$$$\:\:\:\:{prove}\:\:{E}\:{is}\:{open}\:{in}\:{Y}\:{if}\:{and}\:\:{only}\:{if} \\ $$$$\:\:\:\:\:\:\:\exists\:{G}\:\left({open}\:{set}\:\right)\:{in}\:{X}\:\:{such}\:{that} \\ $$$$\:\:\:\:\:\:\:\:\:{E}\:=\:{G}\:\cap\:{Y}\:\:\:….\:\left({mathematical}\:{analysis}\:\left({I}\right)\right) \\ $$ Answered by Berbere last updated…
Question Number 206302 by ayietaamos last updated on 11/Apr/24 Answered by A5T last updated on 11/Apr/24 $${k}=\frac{{ln}\left(\frac{{T}}{\mathrm{3}{y}}\right)−\mathrm{1}}{\mathrm{2}} \\ $$ Answered by MATHEMATICSAM last updated on…