Question Number 85224 by mathocean1 last updated on 20/Mar/20 $${solve}\:{in}\:\mathbb{N} \\ $$$$\left({n}−\mathrm{4}\right)!=\mathrm{42}\left({n}−\mathrm{2}\right)! \\ $$ Commented by jagoll last updated on 20/Mar/20 $$\left(\mathrm{n}−\mathrm{4}\right)!\:=\:\mathrm{42}\left(\mathrm{n}−\mathrm{2}\right)\left(\mathrm{n}−\mathrm{3}\right)\left(\mathrm{n}−\mathrm{4}\right)! \\ $$$$\mathrm{1}\:=\:\mathrm{42}\left(\mathrm{n}−\mathrm{2}\right)\left(\mathrm{n}−\mathrm{3}\right) \\…
Question Number 19687 by Tinkutara last updated on 14/Aug/17 $$\mathrm{Let}\:{z}_{\mathrm{1}} ,\:{z}_{\mathrm{2}} ,\:{z}_{\mathrm{3}} \:\mathrm{be}\:\mathrm{three}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{an} \\ $$$$\mathrm{equilateral}\:\mathrm{triangle}\:\mathrm{circumscribing}\:\mathrm{the} \\ $$$$\mathrm{circle}\:\mid{z}\mid\:=\:\frac{\mathrm{1}}{\mathrm{2}}.\:\mathrm{If}\:{z}_{\mathrm{1}} \:=\:\frac{\mathrm{1}}{\mathrm{2}}\:+\:\frac{\sqrt{\mathrm{3}}{i}}{\mathrm{2}}\:\mathrm{and}\:{z}_{\mathrm{1}} , \\ $$$${z}_{\mathrm{2}} ,\:{z}_{\mathrm{3}} \:\mathrm{are}\:\mathrm{in}\:\mathrm{anticlockwise}\:\mathrm{sense}\:\mathrm{then}\:{z}_{\mathrm{2}} \:\mathrm{is} \\…
Question Number 85183 by Serlea last updated on 19/Mar/20 $$\mathrm{Hi}\:\mathrm{veterans} \\ $$$$\mathrm{Serlea}\:\left(\mathrm{1}\right) \\ $$$$ \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{last}\:\mathrm{three}\:\mathrm{digits}\:\mathrm{of}: \\ $$$$\mathrm{3005}^{\mathrm{11}} +\mathrm{3005}^{\mathrm{12}} +\mathrm{3005}^{\mathrm{13}} +…+\mathrm{3005}^{\mathrm{3002}} \\ $$$$ \\ $$$$…
Question Number 150719 by mathdanisur last updated on 14/Aug/21 Commented by MJS_new last updated on 15/Aug/21 $$\frac{\mathrm{1}+\sqrt{\mathrm{21}}}{\mathrm{2}}\leqslant{x}+{y}\leqslant\mathrm{1}+\sqrt{\mathrm{11}} \\ $$ Commented by mathdanisur last updated on…
Find-the-sum-of-all-possible-digits-that-comes-at-ten-s-place-for-3-n-where-n-is-any-natural-number-
Question Number 19646 by Tinkutara last updated on 13/Aug/17 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{possible}\:\mathrm{digits}\:\mathrm{that} \\ $$$$\mathrm{comes}\:\mathrm{at}\:\mathrm{ten}'\mathrm{s}\:\mathrm{place}\:\mathrm{for}\:\mathrm{3}^{{n}} \:\mathrm{where}\:{n}\:\mathrm{is} \\ $$$$\mathrm{any}\:\mathrm{natural}\:\mathrm{number}. \\ $$ Commented by Tinkutara last updated on 16/Aug/17 $$\mathrm{help}\:\mathrm{pls}…
Question Number 19643 by Tinkutara last updated on 13/Aug/17 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{real}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\sqrt{\mathrm{17}\:+\:\mathrm{8}{x}\:−\:\mathrm{2}{x}^{\mathrm{2}} }\:+\:\sqrt{\mathrm{4}\:+\:\mathrm{12}{x}\:−\:\mathrm{3}{x}^{\mathrm{2}} }\:=\:{x}^{\mathrm{2}} \\ $$$$−\:\mathrm{4}{x}\:+\:\mathrm{13}. \\ $$ Answered by ajfour last updated on 13/Aug/17…
Question Number 150718 by mathdanisur last updated on 14/Aug/21 $$\mathrm{Calculate}: \\ $$$$\mathrm{Li}_{\mathrm{2}} \left(\boldsymbol{\mathrm{z}}\right)\:+\:\mathrm{Li}_{\mathrm{2}} \left(\mathrm{1}\:-\:\boldsymbol{\mathrm{z}}\right)\:=\:? \\ $$ Answered by Olaf_Thorendsen last updated on 14/Aug/21 $$\mathrm{Li}\left({z}\right)\:=\:−\int_{\mathrm{0}} ^{{z}}…
Question Number 19638 by Tinkutara last updated on 13/Aug/17 $$\mathrm{Let}\:{P}\left({x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{such}\:\mathrm{that} \\ $$$${P}\left(\mathrm{1}\right)\:=\:\mathrm{1},\:{P}\left(\mathrm{2}\right)\:=\:\mathrm{2},\:{P}\left(\mathrm{3}\right)\:=\:\mathrm{3},\:\mathrm{and} \\ $$$${P}\left(\mathrm{4}\right)\:=\:\mathrm{5}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{P}\left(\mathrm{6}\right). \\ $$ Answered by ajfour last updated on 13/Aug/17 $$\mathrm{let}\:\mathrm{P}\left(\mathrm{x}\right)=\mathrm{ax}^{\mathrm{3}} +\mathrm{bx}^{\mathrm{2}}…
Question Number 85176 by jagoll last updated on 19/Mar/20 $$\mathrm{what}\:\mathrm{is}\:\mathrm{range}\: \\ $$$$\mathrm{function}\:\mathrm{y}\:=\:\sqrt{\mathrm{x}−\mathrm{1}}\:+\:\sqrt{\mathrm{5}−\mathrm{x}} \\ $$ Answered by john santu last updated on 19/Mar/20 $$\mathrm{Domain}\::\:\mathrm{x}\:\geqslant\:\mathrm{1}\:\wedge\:\mathrm{x}\:\leqslant\:\mathrm{5}\:\Rightarrow\:\mathrm{1}\:\leqslant\mathrm{x}\leqslant\mathrm{5} \\ $$$$\mathrm{Range}\::\:\mathrm{y}\:'\:=\:\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{x}−\mathrm{1}}}\:−\:\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{5}−\mathrm{x}}}\:=\:\mathrm{0}…
Question Number 150697 by mathdanisur last updated on 14/Aug/21 Answered by dumitrel last updated on 14/Aug/21 $$\mathrm{1011}\leqslant{a}_{{i}} \leqslant\mathrm{2022} \\ $$$$\frac{\mathrm{1}}{\mathrm{2022}}\leqslant\frac{\mathrm{1}}{{b}_{{i}} }\leqslant\frac{\mathrm{1}}{\mathrm{1011}}\Rightarrow \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\leqslant\frac{{a}_{{i}} }{{b}_{{i}} }\leqslant\mathrm{2}\Rightarrow\left(\frac{{a}_{{i}}…