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Category: Number Theory

For-all-n-1-n-Z-prove-that-p-n-4-8-4n-2n-n-1-

Question Number 12209 by tawa last updated on 16/Apr/17 $$\mathrm{For}\:\mathrm{all}\:\mathrm{n}\:\geqslant\:\mathrm{1}\:,\:\:\mathrm{n}\:\in\:\mathrm{Z},\:\:\mathrm{prove}\:\mathrm{that},\: \\ $$$$\mathrm{p}\left(\mathrm{n}\right)\::\:\mathrm{4}\:+\:\mathrm{8}\:+\:…\:+\:\mathrm{4n}\:=\:\mathrm{2n}\left(\mathrm{n}\:+\:\mathrm{1}\right) \\ $$ Commented by tawa last updated on 16/Apr/17 $$\mathrm{please}\:\mathrm{show}\:\mathrm{me}\:\mathrm{full}\:\mathrm{workings}\:\mathrm{sirs}.\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{all}. \\ $$ Answered…

Question-12106

Question Number 12106 by tawa last updated on 13/Apr/17 Answered by FilupS last updated on 13/Apr/17 $$\Sigma{a}_{{n}} \:\mathrm{diverges}\:\mathrm{if}:\:\:\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{a}_{{n}} \neq\mathrm{0} \\ $$$$\Sigma{ca}_{{n}} ={c}\Sigma{a}_{{n}} \\ $$$$\therefore{c}\Sigma{a}_{{n}}…

Solve-by-mathematical-induction-that-1-1-1-2-1-1-2-3-1-1-2-3-n-2n-n-1-

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

Prove-that-n-n-k-1-1-p-k-n-Euler-totient-function-

Question Number 142880 by Dwaipayan Shikari last updated on 06/Jun/21 $$\:{Prove}\:{that}\:\boldsymbol{\phi}\left({n}\right)={n}\underset{{k}} {\prod}\left(\mathrm{1}−\frac{\mathrm{1}}{{p}_{{k}} }\right)\:\:\phi\left({n}\right):{Euler}\:{totient}\:{function} \\ $$ Answered by Snail last updated on 06/Jun/21 $${I}\:{am}\:{considering}\:{that}\:{u}\:{know}\:\phi\left({n}\right)\:{is}\:{multiplicative} \\ $$$${function}\:{i}.{e}\:\phi\left({ab}\right)=\phi\left({a}\right)\phi\left({b}\right)..{where}\:{a}\:{and}\:{b}\:…

Prove-using-the-density-of-Q-in-R-that-every-real-number-x-is-the-limit-of-a-cauchy-sequence-of-rational-numbers-r-n-n-N-Give-a-sequence-of-irrational-numbers-S-n-such-that-S-n-x-

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

550-11-50-And-here-5-2-5-2-0-2-50-find-another-three-digit-number-which-is-divisible-by-11-and-the-quotient-is-the-sum-of-the-square-of-the-every-digit-of-the-dividend-

Question Number 11664 by Nayon last updated on 29/Mar/17 $$\mathrm{550}\boldsymbol{\div}\mathrm{11}=\mathrm{50}\:{And}\:{here}\:\mathrm{5}^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} +\mathrm{0}^{\mathrm{2}} =\mathrm{50} \\ $$$${find}\:{another}\:{three}\:{digit}\:{number} \\ $$$${which}\:{is}\:{divisible}\:{by}\:\mathrm{11}\:{and}\:{the} \\ $$$${quotient}\:{is}\:{the}\:{sum}\:{of}\:{the}\: \\ $$$$\:{square}\:{of}\:\:{the}\:{every}\:{digit}\:{of}\: \\ $$$${the}\:{dividend}. \\ $$$$…

What-is-the-smallest-positive-integer-x-for-which-1-32-x-10-y-for-some-positive-integer-y-

Question Number 11606 by Joel576 last updated on 29/Mar/17 $$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{positive}\:\mathrm{integer}\:{x}\:\mathrm{for} \\ $$$$\mathrm{which}\:\frac{\mathrm{1}}{\mathrm{32}}\:=\:\frac{{x}}{\mathrm{10}^{{y}} }\:\:\mathrm{for}\:\mathrm{some}\:\mathrm{positive}\:\mathrm{integer}\:{y}\:? \\ $$ 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}}…