Question Number 189126 by Mingma last updated on 12/Mar/23 Answered by mr W last updated on 12/Mar/23 $${if}\:\mathrm{6}{k}\leqslant{n}<\mathrm{6}{k}+\mathrm{2}: \\ $$$$\mathrm{3}{k}+\mathrm{2}{k}+{k}={n}\:\Rightarrow{n}=\mathrm{6}{k}\:\checkmark \\ $$$${if}\:\mathrm{6}{k}+\mathrm{2}\leqslant{n}<\mathrm{6}{k}+\mathrm{3}: \\ $$$$\mathrm{3}{k}+\mathrm{1}+\mathrm{2}{k}+{k}={n}\:\Rightarrow{n}=\mathrm{6}{k}+\mathrm{1}\:\rightarrow{bad} \\…
Question Number 123488 by peter frank last updated on 25/Nov/20 Answered by MJS_new last updated on 25/Nov/20 $${x}=\mathrm{sech}\:{y}\:=\frac{\mathrm{1}}{\mathrm{cosh}\:{y}}=\frac{\mathrm{2e}^{{y}} }{\mathrm{e}^{\mathrm{2}{y}} +\mathrm{1}} \\ $$$$\Rightarrow\:{y}=\mathrm{ln}\:\frac{\mathrm{1}+\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{{x}}\:=\mathrm{sech}^{−\mathrm{1}} \:{x} \\…
Question Number 188970 by BaliramKumar last updated on 09/Mar/23 $$\mathrm{LCM}\left({x},\:\mathrm{144},\:\mathrm{150}\right)\:=\:\mathrm{10800}\:{how}\:{many}\:{value}\:{of}\:\:{x}. \\ $$$$ \\ $$ Answered by mr W last updated on 10/Mar/23 $$\mathrm{10800}=\mathrm{2}^{\mathrm{4}} ×\mathrm{3}^{\mathrm{3}} ×\mathrm{5}^{\mathrm{2}}…
Question Number 188881 by mnjuly1970 last updated on 08/Mar/23 $$ \\ $$$$\:\:\:\:\mathrm{Q}\::\:\:\mathrm{the}\:\mathrm{non}−\mathrm{zero}\:\mathrm{vector}\:\overset{\rightarrow} {{a}}\:=\:\left({a}_{\mathrm{1}} \:,\:{a}_{\:\mathrm{2}} \:,\:{a}_{\:\mathrm{3}} \:\right)\:\mathrm{with} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{the}\:\:\mathrm{coordinate}\:\mathrm{axes}\:\mathrm{makes}\:\mathrm{the} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{angles}\:\:,\:\:\alpha\:\:\:,\:\:\beta\:\:\mathrm{and}\:\:\:\gamma\:.\:\:\mathrm{prove} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{that}\:\:\mathrm{the}\:\mathrm{following}\:\mathrm{equality}. \\ $$$$\:\: \\…
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Question Number 57770 by Tawa1 last updated on 11/Apr/19 Answered by Kunal12588 last updated on 11/Apr/19 $$\underset{{k}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}\frac{\mathrm{1}}{{sin}\left(\frac{\pi}{\mathrm{4}}+\frac{\left({k}−\mathrm{1}\right)\pi}{\mathrm{6}}\right){sin}\left(\frac{\pi}{\mathrm{4}}+\frac{{k}\pi}{\mathrm{6}}\right)} \\ $$$${sin}\left(\frac{\pi}{\mathrm{4}}+\frac{\left({k}−\mathrm{1}\right)\pi}{\mathrm{6}}\right){sin}\left(\frac{\pi}{\mathrm{4}}+\frac{{k}\pi}{\mathrm{6}}\right) \\ $$$$={sin}\left(\frac{\mathrm{3}\pi+\mathrm{2}\left({k}−\mathrm{1}\right)\pi}{\mathrm{12}}\right){sin}\left(\frac{\mathrm{3}\pi+\mathrm{2}{k}\pi}{\mathrm{12}}\right) \\ $$$$={sin}\left(\frac{\left(\mathrm{2}{k}+\mathrm{1}\right)\pi}{\mathrm{12}}\right){sin}\left(\frac{\left(\mathrm{2}{k}+\mathrm{3}\right)\pi}{\mathrm{12}}\right)…
Question Number 123304 by mordo last updated on 24/Nov/20 $${gcd}\left({a};{b}\right)+{lcm}\left({a};{b}\right)=\mathrm{111} \\ $$$${find}\:{a}\:{and}\:{b} \\ $$ Commented by mr W last updated on 24/Nov/20 $${a}=\mathrm{1} \\ $$$${b}=\mathrm{110}…
Question Number 188796 by Rupesh123 last updated on 07/Mar/23 Commented by mr W last updated on 07/Mar/23 $${consider}\:{your}\:{question}: \\ $$$${f}\left({n}\right)\:{should}\:{be}\:{divisible}\:{by}\:{n}.\:{but} \\ $$$${the}\:{examples}\:{you}\:{gave}\:{are}\:{not} \\ $$$${fibonacci}\:{series}\:{and}\:{are}\:{not}\:{divisible} \\…
Question Number 188722 by Rupesh123 last updated on 06/Mar/23 Commented by Rupesh123 last updated on 06/Mar/23 Find the number of rational terms in the binomial expansion of Answered by BaliramKumar last updated on 06/Mar/23 $$\overset{\mathrm{100}}…
Question Number 188564 by BaliramKumar last updated on 03/Mar/23 $$\mathrm{1}×\mathrm{2}+\mathrm{2}×\mathrm{3}+\mathrm{3}×\mathrm{4}+……………+\mathrm{24}×\mathrm{25}\:=\:? \\ $$ Answered by universe last updated on 03/Mar/23 $$\left(\mathrm{1}^{\mathrm{2}} +\mathrm{1}\right)+\left(\mathrm{2}^{\mathrm{2}} +\mathrm{2}\right)+\left(\mathrm{3}^{\mathrm{2}} +\mathrm{3}\right)+……\left(\mathrm{24}^{\mathrm{2}} +\mathrm{24}\right) \\…