Category: Number Theory

Question Number 34186 by Joel578 last updated on 02/May/18 $$\mathrm{Let}{x}_1=0,x_2=1\mathrm{and}{x}_n=\frac{1}{2}(x_{n-1}+x_{n-2})$$$$\mathrm{Show}\mathrm{that}$$$$x_n=\frac{2^{n-1}+{(-1)}^n}{3.2^{n-2}}$$…

Question Number 164940 by amin96 last updated on 23/Jan/22 $$---------$$$$1!-2!+3!-4!+5!-\dots -14!+15!=?$$$$$$$$----------\mathit{b}\mathit{y}\mathit{M}.\mathit{A}$$ Commented by MJS_new last updated on 24/Jan/22…

Question Number 33463 by NECx last updated on 17/Apr/18 $$Findthepricipalandordinary$$$$argumentofz=\frac{i}{-2-2i}$$ Commented by abdo imad last updated on 17/Apr/18 $$z=\frac{-i}{2(1+i)}=\frac{-i(1-i)}{4}=\frac{1-i}{4}\Rightarrow \mid z\mid =\frac{1}{4}\mid 1-i\mid =\frac{\sqrt{2}}{4}$$$${z}\:=\frac{\mathrm{1}}{\mathrm{4}}\:−\frac{{i}}{\mathrm{4}}\:=\frac{\sqrt{\mathrm{2}}}{\mathrm{4}}\left(\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:−\frac{{i}}{\:\sqrt{\mathrm{2}}}\right)\:=\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\:{e}^{−{i}\frac{\pi}{\mathrm{4}}}…

Question Number 33323 by Rasheed.Sindhi last updated on 14/Apr/18 $$\text{You can't use 'macro parameter character \#' in math mode}$$$$\mathrm{If}{\mathrm{A}}_1,{\mathrm{A}}_2,\dots {\mathrm{A}}_{\mathrm{n}}\mathrm{are}\mathrm{n}\mathrm{points}\mathrm{with}\mathrm{integer}$$$$\mathrm{coordinates}\mathrm{of}\mathrm{a}\mathrm{plane}\mathrm{such}\mathrm{that}\mathrm{every}\mathrm{triangle}$$$$\mathrm{whose}\mathrm{vertices}\mathrm{are}\mathrm{any}\mathrm{three}\mathrm{of}\mathrm{the}\mathrm{above}$$$$\mathrm{points}\mathrm{has}\mathrm{its}\mathrm{centroid}\mathrm{with}\mathrm{at}\mathrm{least}\mathrm{one}$$$$\mathrm{non}-\mathrm{integer}\mathrm{coordinate}.\mathrm{Find}\mathrm{the}\mathrm{maximum}$$$$\mathrm{possible}\:\mathrm{n}.\:\:\:…

Question Number 98602 by bemath last updated on 15/Jun/20 $$\mathrm{find}\mathrm{integral}\mathrm{solution}$$$$\mathrm{of}{\mathrm{y}}^2={\mathrm{x}}^3+1$$ Commented by bemath last updated on 15/Jun/20 $$\mathrm{oo}\:\mathrm{yes}\:\mathrm{sir}.\:\mathrm{thank}\:\mathrm{you} \…