Question Number 214329 by malwan last updated on 05/Dec/24 $${what}\:{is}\:{the}\:{coefficient}\:{of} \\ $$$${x}^{\mathrm{50}\:\:} \:{in} \\ $$$$\left(\mathrm{1}+\mathrm{2}{x}+\mathrm{3}{x}^{\mathrm{2}} +…+\mathrm{101}{x}^{\mathrm{100}} \right)\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +…+{x}^{\mathrm{25}} \right) \\ $$ Answered by mr W…
Question Number 214326 by mathlove last updated on 05/Dec/24 $$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\sqrt[{\mathrm{4}}]{{x}}−\sqrt[{\mathrm{6}}]{{x}}}{\:\sqrt[{\mathrm{4}}]{{x}}+\sqrt[{\mathrm{6}}]{{x}}}=? \\ $$ Answered by malwan last updated on 05/Dec/24 $${put}\:{y}={x}^{\mathrm{12}} \\ $$$$\underset{{y}\rightarrow\infty} {{lim}}\:\frac{{y}^{\mathrm{3}} −{y}^{\mathrm{2}}…
Question Number 214321 by MathematicsExpert last updated on 05/Dec/24 $$\mathrm{15}{x}\left(\frac{\mathrm{6}{x}}{\mathrm{10}}+\frac{\mathrm{12}{x}}{\mathrm{20}}+\frac{\mathrm{15}{x}}{\mathrm{30}}+…+\frac{\mathrm{60}{x}}{\mathrm{100}}\right)=\mathrm{160} \\ $$$$\mathrm{solve}\:\mathrm{for}\:{x} \\ $$ Answered by MATHEMATICSAM last updated on 06/Dec/24 $$\mathrm{15}{x}\left(\frac{\mathrm{6}{x}}{\mathrm{10}}\:+\:\frac{\mathrm{12}{x}}{\mathrm{20}}\:+\:\frac{\mathrm{15}{x}}{\mathrm{30}}\:+\:…\:+\:\frac{\mathrm{60}{x}}{\mathrm{100}}\right)\:=\:\mathrm{160} \\ $$$$\Rightarrow\:\mathrm{15}{x}\left(\frac{\mathrm{3}{x}}{\mathrm{5}}\:+\:\frac{\mathrm{3}{x}}{\mathrm{5}}\:+\:\frac{\mathrm{3}{x}}{\mathrm{5}}\:+\:…\:+\:\frac{\mathrm{3}{x}}{\mathrm{5}}\right)\:=\:\mathrm{160} \\…
Question Number 214322 by MathematicsExpert last updated on 05/Dec/24 $$\mathrm{find}\:\mathrm{the}\:\mathrm{errors} \\ $$$$\mathrm{7}{x}+\mathrm{4}=\mathrm{10}{x}−\mathrm{6} \\ $$$$\mathrm{7}{x}+\mathrm{4}−\mathrm{10}{x}=\mathrm{10}{x}−\mathrm{6}−\mathrm{10}{x} \\ $$$$−\mathrm{3}{x}+\mathrm{4}=−\mathrm{6} \\ $$$$−\mathrm{3}{x}+\mathrm{4}+\mathrm{6}=−\mathrm{6}+\mathrm{6} \\ $$$$−\mathrm{3}{x}+\mathrm{10}=\mathrm{0} \\ $$$$−\mathrm{3}{x}+\mathrm{10}−\mathrm{10}=\mathrm{0}−\mathrm{10} \\ $$$$−\mathrm{3}{x}=−\mathrm{10} \\…
Question Number 214301 by efronzo1 last updated on 04/Dec/24 $$\:\:\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\:\mathrm{sin}\:^{\mathrm{2}} \left(\mathrm{sin}\:\mathrm{x}\right)+\:\mathrm{cos}\:^{\mathrm{2}} \left(\mathrm{cos}\:\mathrm{x}\right)\:\mathrm{dx}\:=? \\ $$ Commented by Frix last updated on 04/Dec/24 $$\mathrm{Should}\:\mathrm{be}\:\frac{\pi}{\mathrm{2}} \\…
Question Number 214302 by universe last updated on 04/Dec/24 $$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{{n}\left(\mathrm{4}{n}−\mathrm{1}\right)^{\mathrm{2}} }=\:? \\ $$ Answered by MrGaster last updated on 24/Dec/24 $$\left(\mathrm{4}{n}−\mathrm{1}\right)^{\mathrm{2}} =\mathrm{16}{n}^{\mathrm{2}} −\mathrm{8}{n}+\mathrm{1}…
Question Number 214297 by ajfour last updated on 04/Dec/24 Commented by ajfour last updated on 04/Dec/24 $${the}\:{tangents}\:{are}\:{at}\:{right}\:{angles}\:{to} \\ $$$${one}\:{another}\:{and}\:{corner}\:{maynot}\:{be} \\ $$$${at}\:{semicircle}\:{centre}. \\ $$ Commented by…
Question Number 214293 by muallimRiyoziyot last updated on 04/Dec/24 $${P}\left({x}\right)\:\:\:\:\:\:\vdots\left({x}^{\mathrm{2}} +\mathrm{3}\right)\:\:\:\:\:{mod}\left(\mathrm{5}{x}−\mathrm{1}\right) \\ $$$${P}\left({x}\right)\:\:\:\:\:\:\vdots\left({x}−\mathrm{2}\right)\:\:\:\:\:\:\:\:\:{mod}\left(\mathrm{16}\right) \\ $$$${P}\left({x}\right)\:\:\:\:\:\:\vdots\left({x}^{\mathrm{2}} +\mathrm{3}\right)\left({x}−\mathrm{2}\right)\:\:\:{mod}\left(?\right) \\ $$ Answered by MrGaster last updated on 24/Dec/24…
Question Number 214310 by malwan last updated on 04/Dec/24 $$\frac{\mathrm{1}}{\mathrm{2}!}\:+\:\frac{\mathrm{2}}{\mathrm{3}!}\:+\:\frac{\mathrm{3}}{\mathrm{4}!}\:+\:…\:+\:\frac{\mathrm{99}}{\mathrm{100}!} \\ $$ Answered by mr W last updated on 05/Dec/24 $$\frac{{n}−\mathrm{1}}{{n}!}=\frac{{n}}{{n}!}−\frac{\mathrm{1}}{{n}!}=\frac{\mathrm{1}}{\left({n}−\mathrm{1}\right)!}−\frac{\mathrm{1}}{{n}!} \\ $$$$ \\ $$$${sum}=\left(\frac{\mathrm{1}}{\mathrm{1}!}−\frac{\mathrm{1}}{\mathrm{2}!}\right)+\left(\frac{\mathrm{1}}{\mathrm{2}!}−\frac{\mathrm{1}}{\mathrm{3}!}\right)+\left(\frac{\mathrm{1}}{\mathrm{3}!}−\frac{\mathrm{1}}{\mathrm{4}!}\right)+…+\left(\frac{\mathrm{1}}{\mathrm{99}!}−\frac{\mathrm{1}}{\mathrm{100}!}\right)…
Question Number 214280 by Karleepsingh1438 last updated on 03/Dec/24 Commented by JuniorKepler last updated on 03/Dec/24 $$\frac{\mathrm{25}\:×\:\mathrm{5}^{\mathrm{2}} \:×\:{t}^{\mathrm{8}} \:}{\mathrm{10}^{\mathrm{3}} \:×\:{t}^{\mathrm{4}} }\:=\:\frac{\mathrm{25}\:×\:\mathrm{25}\:×\:{t}^{\mathrm{4}} }{\mathrm{1000}} \\ $$$$=\:\frac{\mathrm{5}{t}^{\mathrm{4}} }{\mathrm{4}}…