Question Number 162552 by amin96 last updated on 30/Dec/21 $$\boldsymbol{\alpha}_{\mathrm{1}} <\boldsymbol{\alpha}_{\mathrm{2}} <\boldsymbol{\alpha}_{\mathrm{3}} <\ldots<\boldsymbol{\alpha}_{{k}} \\ $$$$\frac{\mathrm{2}^{\mathrm{289}} +\mathrm{1}}{\mathrm{2}^{\mathrm{17}} +\mathrm{1}}=\mathrm{2}^{\boldsymbol{\alpha}_{\mathrm{1}} } +\mathrm{2}^{\boldsymbol{\alpha}_{\mathrm{2}} } +\ldots+\mathrm{2}^{\boldsymbol{\alpha}_{{k}} } \:\:\:\:\:\:\:\boldsymbol{\mathrm{k}}=? \\ $$$$…
Question Number 31485 by mondodotto@gmail.com last updated on 09/Mar/18 Answered by ajfour last updated on 09/Mar/18 $$\left({i}\right)\:\:\:{let}\:\:\:\:{y}−\mathrm{2}\:=\:{t}\left({x}−\mathrm{3}\right) \\ $$$$\Rightarrow\:\:\:\:\frac{{dy}}{{dx}}=\left({x}−\mathrm{3}\right)\frac{{dt}}{{dx}}+{t} \\ $$$${Then}\:\:{diff}.\:{eq}.\:{becomes} \\ $$$$\:\:\:\:\:\:\:\:\left({x}−\mathrm{3}\right)\frac{{dt}}{{dx}}+{t}\:=\:\frac{{t}\left({x}−\mathrm{3}\right)}{\left({t}+\mathrm{1}\right)\left({x}−\mathrm{3}\right)} \\ $$$$\Rightarrow\:\:\left({x}−\mathrm{3}\right)\frac{{dt}}{{dx}}\:+{t}\:=\frac{{t}}{{t}+\mathrm{1}}…
Question Number 97019 by bshahid010@gmail.com last updated on 06/Jun/20 Commented by PRITHWISH SEN 2 last updated on 06/Jun/20 $$\left(\mathrm{2}+\mathrm{2}.\mathrm{8}+\mathrm{2}.\mathrm{8}^{\mathrm{2}} +….+\mathrm{2}.\mathrm{8}^{\mathrm{29}} \right)+\left(\mathrm{4}+\mathrm{4}.\mathrm{8}+\mathrm{4}.\mathrm{8}^{\mathrm{2}} +…+\mathrm{4}.\mathrm{8}^{\mathrm{29}} \right) \\ $$$$−\left(\mathrm{8}+\mathrm{8}.\mathrm{8}+\mathrm{8}.\mathrm{8}^{\mathrm{2}}…
Question Number 162533 by mnjuly1970 last updated on 30/Dec/21 Answered by Ar Brandon last updated on 30/Dec/21 $${x}^{\mathrm{2}} +\mathrm{2}{x}−\mathrm{2}=\mathrm{0} \\ $$$$\alpha+\beta=−\mathrm{2} \\ $$$$\alpha\beta=−\mathrm{2} \\ $$$$\frac{\alpha^{\mathrm{4}}…
Question Number 31456 by rahul 19 last updated on 08/Mar/18 $$\mathbb{F}{ind}\:{sum}\:{of} \\ $$$${S}=\:\frac{\mathrm{2}}{\mathrm{3}}\:+\:\frac{\mathrm{4}}{\mathrm{3}^{\mathrm{2}} }\:+\:\frac{\mathrm{6}}{\mathrm{3}^{\mathrm{3}} }\:+\:\frac{\mathrm{8}}{\mathrm{3}^{\mathrm{4}} }\:+……+\infty\:? \\ $$ Commented by MJS last updated on 08/Mar/18…
Question Number 162521 by HongKing last updated on 30/Dec/21 $$\mathrm{For}\:\mathrm{every}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{number}\:\boldsymbol{\mathrm{x}}\:,\:\mathrm{let} \\ $$$$\mathrm{g}\left(\mathrm{x}\right)\:=\underset{\boldsymbol{\mathrm{r}}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\left(\mathrm{x}+\mathrm{1}\right)^{\boldsymbol{\mathrm{r}}+\mathrm{1}} \:-\:\mathrm{x}^{\boldsymbol{\mathrm{r}}+\mathrm{1}} \right)^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{r}}}} \\ $$$$\mathrm{Find}:\:\:\underset{\boldsymbol{\mathrm{x}}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{g}\left(\mathrm{x}\right)}{\mathrm{x}} \\ $$ Terms of Service Privacy Policy…
Question Number 162520 by HongKing last updated on 30/Dec/21 $$\mathrm{Find}: \\ $$$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\boldsymbol{\pi}} {\int}}\:\left(\frac{\mathrm{x}\:\mathrm{cos}\:\mathrm{x}}{\mathrm{1}\:+\:\mathrm{sin}\:\mathrm{x}}\right)^{\mathrm{2}} \mathrm{dx}\: \\ $$ Answered by Ar Brandon last updated on 30/Dec/21…
Question Number 162522 by HongKing last updated on 30/Dec/21 $$\mathrm{Determine}\:\mathrm{all}\:\mathrm{positive}\:\mathrm{integers} \\ $$$$\boldsymbol{\mathrm{N}}\:\mathrm{which}\:\mathrm{the}\:\mathrm{sphere} \\ $$$$\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \:+\:\mathrm{z}^{\mathrm{2}} \:=\:\mathrm{N} \\ $$$$\mathrm{has}\:\mathrm{an}\:\mathrm{inseribed}\:\mathrm{regular}\:\mathrm{tetrahedron} \\ $$$$\mathrm{whose}\:\mathrm{vertices}\:\mathrm{have}\:\mathrm{integer}\:\mathrm{coordinates} \\ $$ Answered by…
Question Number 162506 by amin96 last updated on 29/Dec/21 Commented by MJS_new last updated on 30/Dec/21 $$\mathrm{no}\:\mathrm{solution}\:\in\mathbb{C} \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 96951 by aurpeyz last updated on 05/Jun/20 $${prove}\:{that}\:\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}−\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{5}}+…+\frac{−\mathrm{1}^{{n}−\mathrm{1}} }{{n}}\:\:{is}\:{always}\:{positive} \\ $$$$ \\ $$ Commented by mr W last updated on 05/Jun/20 $${how}\:{many}\:{times}\:{do}\:{you}\:{want}\:{to}\:{post} \\…