Question Number 19435 by Tinkutara last updated on 11/Aug/17 $$\mathrm{If}\:\alpha\:=\:\mathrm{cos}\:\frac{\mathrm{2}\pi}{\mathrm{5}}\:+\:{i}\:\mathrm{sin}\:\frac{\mathrm{2}\pi}{\mathrm{5}}\:,\:\mathrm{then}\:\mathrm{find}\:\mathrm{the} \\ $$$$\mathrm{value}\:\mathrm{of}\:\alpha\:+\:\alpha^{\mathrm{2}} \:+\:\alpha^{\mathrm{3}} \:+\:\alpha^{\mathrm{4}} . \\ $$ Answered by ajfour last updated on 11/Aug/17 $$\underset{\mathrm{r}=\mathrm{1}}…
Question Number 19434 by Tinkutara last updated on 11/Aug/17 $$\mathrm{If}\:\left(\frac{\mathrm{1}\:+\:{i}\sqrt{\mathrm{3}}}{\mathrm{1}\:−\:{i}\sqrt{\mathrm{3}}}\right)^{{n}} \:\mathrm{is}\:\mathrm{an}\:\mathrm{integer},\:\mathrm{then}\:{n}\:\mathrm{is} \\ $$ Answered by ajfour last updated on 11/Aug/17 $$\left(\frac{\mathrm{1}+{i}\sqrt{\mathrm{3}}}{\mathrm{1}−{i}\sqrt{\mathrm{3}}}\right)^{\mathrm{n}} =\left(\frac{\mathrm{e}^{{i}\pi/\mathrm{6}} }{\mathrm{e}^{−{i}\pi/\mathrm{6}} }\right)^{\mathrm{n}} =\left[\mathrm{x}\right]…
Question Number 19433 by Tinkutara last updated on 11/Aug/17 $$\frac{\sqrt{\mathrm{5}\:+\:\mathrm{12}{i}}\:+\:\sqrt{\mathrm{5}\:−\:\mathrm{12}{i}}}{\:\sqrt{\mathrm{5}\:+\:\mathrm{12}{i}}\:−\:\sqrt{\mathrm{5}\:−\:\mathrm{12}{i}}}\:= \\ $$ Answered by ajfour last updated on 11/Aug/17 $$=\frac{\left(\sqrt{\mathrm{5}+\mathrm{12i}}+\sqrt{\mathrm{5}−\mathrm{12i}}\right)^{\mathrm{2}} }{\mathrm{24i}} \\ $$$$=\frac{\mathrm{10}+\mathrm{2}\sqrt{\mathrm{25}+\mathrm{144}}}{\mathrm{24i}}=\frac{\mathrm{3}}{\mathrm{2i}}=−\frac{\mathrm{3}}{\mathrm{2}}\mathrm{i}\:. \\ $$…
Question Number 84969 by subhankar10 last updated on 18/Mar/20 $$\mathrm{let}\:\mathrm{c}\:\mathrm{is}\:\mathrm{a}\:\mathrm{constant}\:\mathrm{vector}\:\mathrm{and}\:\overset{\rightarrow} {\mathrm{r}}=\mathrm{x}\hat {\mathrm{i}}+\mathrm{y}\hat {\mathrm{j}}+\mathrm{z}\hat {\mathrm{k}}\:\mathrm{then}\:\mathrm{proved}\:\mathrm{that}\:\mathrm{grad}\:\mid\mathrm{c}×\overset{\rightarrow} {\mathrm{r}}\mid^{\mathrm{n}} =\mathrm{n}\mid\mathrm{c}×\overset{\rightarrow} {\mathrm{r}}\mid^{\mathrm{n}−\mathrm{2}} \mathrm{c}×\left(\overset{\rightarrow} {\mathrm{r}}×\mathrm{c}\right). \\ $$ Terms of Service Privacy…
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Question Number 150500 by EnterUsername last updated on 13/Aug/21 $$\mathrm{Find}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{first}-\mathrm{n}\:\mathrm{terms}\:\mathrm{in} \\ $$$${u}_{{n}} =\mathrm{4},\:\mathrm{9},\:\mathrm{15},\:\mathrm{23},\:\mathrm{35},\:\mathrm{55},\:\mathrm{91},\:\mathrm{159},… \\ $$ Commented by CAIMAN last updated on 13/Aug/21 Un=2^n-1+4n -1 Answered by…
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Question Number 84960 by jagoll last updated on 18/Mar/20 Answered by jagoll last updated on 18/Mar/20 Answered by mr W last updated on 18/Mar/20 $$\frac{{x}_{\mathrm{1}}…
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Question Number 84956 by M±th+et£s last updated on 17/Mar/20 $${show}\:{that}\: \\ $$$$\int_{\mathrm{0}} ^{+\infty} \frac{\mathrm{1}}{{x}^{\mathrm{4}} +\mathrm{2}{x}^{\mathrm{2}} {cos}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)+\mathrm{1}}\:{dx}=\frac{\pi}{\mathrm{2}\phi} \\ $$ Commented by mathmax by abdo last updated…