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Category: Arithmetic

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Question Number 3204 by Filup last updated on 07/Dec/15 $$\underset{{i}=\mathrm{1}} {\overset{{n}} {\prod}}{i}!=\mathrm{1}×\left(\mathrm{1}×\mathrm{2}\right)×…×\left(\mathrm{1}×\mathrm{2}×…×{n}\right) \\ $$$$=\mathrm{1}^{{n}} \mathrm{2}^{{n}−\mathrm{1}} \mathrm{3}^{{n}−\mathrm{2}} …\left({n}−\mathrm{1}\right)^{\mathrm{2}} {n} \\ $$$$\therefore\:\underset{{i}=\mathrm{1}} {\overset{{n}} {\prod}}{i}!=\underset{{i}=\mathrm{1}} {\overset{{n}} {\prod}}\left({n}−{i}+\mathrm{1}\right)^{{i}} \\…

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Question Number 3198 by Filup last updated on 07/Dec/15 $$\underset{{i}={a}} {\overset{{n}} {\prod}}{i}=?\:\:\:\:\:\mathrm{for}\:\mid{i}\mid<\mathrm{1},\:\:{a}\in\mathbb{R} \\ $$ Commented by Filup last updated on 07/Dec/15 $$=\frac{\mathrm{1}}{{a}}×\frac{\mathrm{1}}{{a}+\mathrm{1}}×…×\frac{\mathrm{1}}{{n}} \\ $$$$=\frac{\mathrm{1}}{\left(\frac{{n}!}{\left({a}−\mathrm{1}\right)!}\right)} \\…

Question-68710

Question Number 68710 by peter frank last updated on 15/Sep/19 Commented by Prithwish sen last updated on 15/Sep/19 $$\mathrm{x}+\mathrm{y}=\mathrm{0}…….\left(\mathrm{i}\right) \\ $$$$\mathrm{x}−\mathrm{y}=−\mathrm{4}……\left(\mathrm{ii}\right) \\ $$$$\mathrm{solving}\:\mathrm{x}=−\mathrm{2},\mathrm{y}=\mathrm{2} \\ $$$$\therefore\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{center}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{circle}}\:\boldsymbol{\mathrm{is}}\:\left(−\mathrm{2},\mathrm{2}\right)…

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Question Number 3105 by prakash jain last updated on 04/Dec/15 $$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{next}\:\mathrm{term}\:\mathrm{in}\:\mathrm{the}\:\mathrm{below}\:\mathrm{sequence} \\ $$$$\mathrm{1},\mathrm{3},\mathrm{9},\mathrm{27},\mathrm{81},\mathrm{243},\mathrm{729},\:\mathrm{2123},\:\mathrm{5857},? \\ $$ Commented by Filup last updated on 04/Dec/15 $$\mathrm{3}^{\mathrm{10}} ? \\…

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Question Number 3077 by Filup last updated on 04/Dec/15 $$\mathrm{Read}\:\mathrm{the}\:\mathrm{following}: \\ $$$$ \\ $$$${S}=\underset{{i}=\mathrm{1}} {\overset{\infty} {\sum}}{n}^{{i}!} \\ $$$${S}={n}^{\mathrm{1}} +{n}^{\mathrm{1}×\mathrm{2}} +{n}^{\mathrm{1}×\mathrm{2}×\mathrm{3}} +… \\ $$$${S}={n}^{\mathrm{1}} \left(\mathrm{1}+{n}^{\mathrm{2}} +{n}^{\mathrm{2}×\mathrm{3}}…

Question-68589

Question Number 68589 by peter frank last updated on 13/Sep/19 Answered by $@ty@m123 last updated on 14/Sep/19 $$\underset{{k}=\mathrm{2}} {\overset{{n}} {\sum}}\:\:\frac{\mathrm{1}}{{n}\left({n}+\mathrm{1}\right)} \\ $$$$=\underset{{k}=\mathrm{2}} {\overset{{n}} {\sum}}\:\left(\:\frac{\mathrm{1}}{{n}}−\frac{\mathrm{1}}{{n}+\mathrm{1}}\right) \\…

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Question Number 3052 by Filup last updated on 04/Dec/15 $$\mathrm{Ever}\:\mathrm{used}\:\mathrm{Arrow}\:\mathrm{Notation}? \\ $$$$ \\ $$$$\mathrm{if}\:\mathrm{my}\:\mathrm{memory}\:\mathrm{is}\:\mathrm{correct}: \\ $$$$\mathrm{3}\uparrow\mathrm{3}=\mathrm{3}×\mathrm{3}×\mathrm{3}=\mathrm{3}^{\mathrm{3}} \\ $$$$\mathrm{3}\uparrow\uparrow\mathrm{3}=\left(\mathrm{3}\uparrow\mathrm{3}\right)\uparrow\mathrm{3}=\mathrm{3}^{\mathrm{3}} ×\mathrm{3}^{\mathrm{3}} ×\mathrm{3}^{\mathrm{3}} \\ $$$$\mathrm{etc}. \\ $$ Commented…