Question Number 201430 by hardmath last updated on 06/Dec/23 $$\mathrm{Find}: \\ $$$$\frac{\mathrm{2}}{\mathrm{35}}\:+\:\frac{\mathrm{2}}{\mathrm{63}}\:+\:\frac{\mathrm{2}}{\mathrm{99}}\:+\:\frac{\mathrm{2}}{\mathrm{143}}\:=\:? \\ $$ Answered by Rasheed.Sindhi last updated on 06/Dec/23 $$\mathrm{2}\left(\frac{\mathrm{1}}{\mathrm{6}^{\mathrm{2}} −\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{8}^{\mathrm{2}} −\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{10}^{\mathrm{2}} −\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{12}^{\mathrm{2}}…
Question Number 201463 by hardmath last updated on 06/Dec/23 $$\mathrm{a}\:=\:\mathrm{constant}\:\mathrm{number}: \\ $$$$\mathrm{if}\:\:\:\int\mathrm{x}\:\centerdot\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:=\:\mathrm{x}^{\mathrm{3}} \:-\:\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{4x}\:-\:\frac{\mathrm{a}}{\mathrm{5}} \\ $$$$\mathrm{find}\:\:\:\mathrm{f}\left(\mathrm{2}\right)\:=\:? \\ $$ Answered by mr W last updated on…
Question Number 201441 by dimentri last updated on 06/Dec/23 $${Let}\:{f}\left({x}\right)\:{and}\:{g}\left({x}\right)\:{be}\:{given}\:{by}\: \\ $$$$\:{f}\left({x}\right)=\:\frac{\mathrm{1}}{{x}}\:+\frac{\mathrm{1}}{{x}−\mathrm{2}}\:+\frac{\mathrm{1}}{{x}−\mathrm{4}}\:+\:…\:+\frac{\mathrm{1}}{{x}−\mathrm{2018}} \\ $$$$\:{and}\: \\ $$$$\:\:{g}\left({x}\right)=\frac{\mathrm{1}}{{x}−\mathrm{1}}\:+\frac{\mathrm{1}}{{x}−\mathrm{3}}\:+\frac{\mathrm{1}}{{x}−\mathrm{5}}\:+…+\:\frac{\mathrm{1}}{{x}−\mathrm{2017}}. \\ $$$$\:\:{Prove}\:{that}\:\:\mid\:{f}\left({x}\right)−{g}\left({x}\right)\mid\:>\mathrm{2} \\ $$$$\:\:{for}\:{any}\:{non}−{integer}\:{real}\:{number} \\ $$$$\:\:{x}\:{satisfying}\:\mathrm{0}<{x}<\mathrm{2018}.\: \\ $$ Answered…
Question Number 201343 by gabi last updated on 05/Dec/23 Answered by aleks041103 last updated on 05/Dec/23 $${A}^{\mathrm{3}} +{A}^{\mathrm{2}} +{A}=\mathrm{0} \\ $$$$\lambda\:{is}\:{eigenvalue} \\ $$$$\Rightarrow\lambda^{\mathrm{3}} +\lambda^{\mathrm{2}} +\lambda=\lambda\left(\lambda^{\mathrm{2}}…
Question Number 201388 by MrGHK last updated on 05/Dec/23 Answered by witcher3 last updated on 05/Dec/23 $$=\underset{\mathrm{n}\geqslant\mathrm{1}} {\sum}\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\mathrm{n}}\underset{\mathrm{m}\geqslant\mathrm{0}} {\sum}\int_{\mathrm{0}} ^{\mathrm{1}} \left(−\mathrm{1}\right)^{\mathrm{m}} \mathrm{x}^{\mathrm{n}+\mathrm{2m}} \mathrm{dx} \\…
Question Number 201408 by Ari last updated on 05/Dec/23 Answered by mr W last updated on 05/Dec/23 $$=\frac{\mathrm{1}×\mathrm{3}}{\mathrm{2}^{\mathrm{2}} }×\frac{\mathrm{2}×\mathrm{4}}{\mathrm{3}^{\mathrm{2}} }×\frac{\mathrm{3}×\mathrm{5}}{\mathrm{4}^{\mathrm{2}} }×…×\frac{\mathrm{98}×\mathrm{100}}{\mathrm{99}^{\mathrm{2}} } \\ $$$$=\frac{\mathrm{1}×\cancel{\mathrm{3}}}{\mathrm{2}^{\cancel{\mathrm{2}}} }×\frac{\cancel{\mathrm{2}}×\cancel{\mathrm{4}}}{\cancel{\mathrm{3}^{\mathrm{2}}…
Question Number 201302 by hardmath last updated on 03/Dec/23 $$\begin{cases}{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{−\mathrm{2}} \:=\:\mathrm{4}}\\{\mathrm{x}^{−\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \:=\:\mathrm{5}}\end{cases}\:\:\:\:\:\mathrm{find}:\:\:\frac{\mathrm{y}}{\mathrm{x}}\:=\:? \\ $$ Answered by witcher3 last updated on 03/Dec/23 $$\left(\mathrm{1}\right)\ast\left(\mathrm{2}\right)\Leftrightarrow\mathrm{2}+\left(\mathrm{xy}\right)^{\mathrm{2}} +\frac{\mathrm{1}}{\left(\mathrm{xy}\right)^{\mathrm{2}}…
Question Number 201282 by Supro last updated on 03/Dec/23 Answered by MM42 last updated on 03/Dec/23 $${x}=\frac{\begin{vmatrix}{{a}^{\mathrm{2}} \:\:\:\:\:\:\:\:\:\:{b}}\\{{ab}\:\:\:\:\:\:−{a}}\end{vmatrix}}{\begin{vmatrix}{{a}\:\:\:\:\:\:\:{b}}\\{{b}\:\:\:\:\:−{a}}\end{vmatrix}}=\frac{−{a}^{\mathrm{3}} −{ab}^{\mathrm{2}} }{−{a}^{\mathrm{2}} −{b}^{\mathrm{2}} }\:={a} \\ $$$${y}=\frac{\begin{vmatrix}{{a}\:\:\:\:\:\:\:\:{a}^{\mathrm{2}} }\\{{b}\:\:\:\:\:\:\:\:{ab}}\end{vmatrix}}{\begin{vmatrix}{{a}\:\:\:\:\:\:\:{b}}\\{{b}\:\:\:\:\:−{a}}\end{vmatrix}}=\frac{{a}^{\mathrm{2}}…
Question Number 201253 by mathlove last updated on 02/Dec/23 $${find}\:{the}\:{sum}\:{of}\:{n}\:{terms}\:{of}\:{the}\:{serice} \\ $$$${s}_{{n}} =\mathrm{5}+\mathrm{11}+\mathrm{19}+\mathrm{29}+\mathrm{41}+………. \\ $$ Answered by Rasheed.Sindhi last updated on 02/Dec/23 $$\left(\mathrm{4}+\mathrm{1}\right)+\left(\mathrm{9}+\mathrm{2}\right)+\left(\mathrm{16}+\mathrm{3}\right)+\left(\mathrm{25}+\mathrm{4}\right)+.. \\ $$$$\left(\mathrm{2}^{\mathrm{2}}…
Question Number 201237 by mnjuly1970 last updated on 02/Dec/23 $$ \\ $$$$\:\:\:\:{If}\:\:,\:\:\:{a}\:\mid\:\mathrm{5}{b}^{\:\mathrm{2}} −\mathrm{10}{b}\:+\mathrm{1}\: \\ $$$$\:\:\:\:\:\Rightarrow\:\:\:\left[\:{a}\:\:,\:\mathrm{5}{b}\:\right]_{\mathrm{lc}{m}} \:=\:\:?\: \\ $$$$ \\ $$$$ \\ $$ Commented by AST…