Category: Number Theory

Question Number 11854 by tawa last updated on 02/Apr/17 $$\mathrm{Solve}\mathrm{by}\mathrm{mathematical}\mathrm{induction}\mathrm{that}$$$$1+\frac{1}{1+2}+\frac{1}{1+2+3}+\dots +\frac{1}{1+2+3+\dots +\mathrm{n}}=\frac{2\mathrm{n}}{\mathrm{n}+1}$$ Answered by sandy_suhendra last updated on 03/Apr/17 $$\mathrm{for}\mathrm{n}=1$$$$\mathrm{1}=\frac{\mathrm{2}.\mathrm{1}}{\mathrm{1}+\mathrm{1}}\:\left(\mathrm{is}\:\mathrm{true}\right) \…

Question Number 11799 by tawa last updated on 01/Apr/17 $$\mathrm{Prove}\mathrm{using}\mathrm{the}\mathrm{density}\mathrm{of}\mathbf{Q}\mathrm{in}\mathbb{R}\mathrm{that}\mathrm{every}\mathrm{real}\mathrm{number}\mathrm{x}\mathrm{is}\mathrm{the}\mathrm{limit}\mathrm{of}$$$$\mathrm{a}\mathrm{cauchy}\mathrm{sequence}\mathrm{of}\mathrm{rational}\mathrm{numbers}{\left({\mathrm{r}}_{\mathrm{n}}\right)}_{\mathrm{n}\in \mathrm{N}}.\mathrm{Give}\mathrm{a}\mathrm{sequence}\mathrm{of}\mathrm{irrational}$$$$\mathrm{numbers}\left({\mathrm{S}}_{\mathrm{n}}\right)\mathrm{such}\mathrm{that}{\mathrm{S}}_{\mathrm{n}}\to \mathrm{x}.$$ Terms of Service Privacy Policy…

Question Number 11606 by Joel576 last updated on 29/Mar/17 $$\mathrm{What}\mathrm{is}\mathrm{the}\mathrm{smallest}\mathrm{positive}\mathrm{integer}x\mathrm{for}$$$$\mathrm{which}\frac{1}{32}=\frac{x}{{10}^y}\mathrm{for}\mathrm{some}\mathrm{positive}\mathrm{integer}y?$$ Answered by mrW1 last updated on 29/Mar/17 $$\mathrm{32}{x}=\mathrm{10}^{{y}} =\left(\mathrm{2}×\mathrm{5}\right)^{{y}} =\mathrm{2}^{{y}}…

Question Number 11321 by Joel576 last updated on 20/Mar/17 $$\mathrm{How}\mathrm{many}\mathrm{solution}\{x,y,z\}\mathrm{that}\mathrm{fulfilled}$$$$x+y+z=99?$$$$x,y,z\in \mathbb{N}$$ Commented by prakash jain last updated on 22/Mar/17 $$\mathrm{Please}\:\mathrm{have}\:\mathrm{a}\:\mathrm{look}\:\mathrm{at}\:\mathrm{stars}\:\mathrm{and}\:\mathrm{bars}…

Question Number 11139 by Joel576 last updated on 13/Mar/17 Commented by Joel576 last updated on 13/Mar/17 $$\mathrm{A}\mathrm{real}\mathrm{number}t,\mathrm{so}\mathrm{there}\mathrm{is}\mathrm{a}\mathrm{three}-\mathrm{ordered}-\mathrm{pair}$$$$\mathrm{solution}\{x,y,z\}\mathrm{that}\mathrm{fulfilled}$$$$x^2+2y^2=3z\mathrm{and}x+y+z=t$$$$\mathrm{Find}\:{t}…