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Category: Arithmetic

Question-81311

Question Number 81311 by M±th+et£s last updated on 11/Feb/20 Commented by mind is power last updated on 11/Feb/20 $$\Gamma\left({n}+{x}\right)=\left({n}−\mathrm{1}+{x}\right)\left({n}−\mathrm{2}+{x}\right)…{x}\Gamma\left({x}\right)\:\: \\ $$$$\Gamma\left({n}+{x}\right)=\Gamma\left({x}\right)\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}\left({n}−{k}+{x}\right) \\ $$$$\Rightarrow\Gamma\left({n}+\mathrm{1}−\frac{{k}}{{n}}\right)=\Gamma\left(\mathrm{1}−\frac{{k}}{{n}}\right)\underset{{j}=\mathrm{1}}…

How-to-calculate-the-last-two-digits-of-2-576-

Question Number 15770 by arnabpapu550@gmail.com last updated on 13/Jun/17 $$\mathrm{How}\:\mathrm{to}\:\mathrm{calculate}\:\mathrm{the}\:\mathrm{last}\:\mathrm{two}\:\mathrm{digits}\:\mathrm{of}\:\:\mathrm{2}^{\mathrm{576}} \\ $$ Answered by Tinkutara last updated on 14/Jun/17 $$\mathrm{2}^{\mathrm{576}} \:=\:\mathrm{2}^{\mathrm{4}×\mathrm{144}} \:\equiv\:\mathrm{6} \\ $$$$\mathrm{Last}\:\mathrm{digit}\:\mathrm{is}\:\mathrm{6}. \\…

Question-81222

Question Number 81222 by M±th+et£s last updated on 10/Feb/20 Answered by mind is power last updated on 10/Feb/20 $$\:\:_{\mathrm{1}} {F}_{{a}} \left(\mathrm{1};\frac{{a}+{b}}{{a}},\frac{{a}+{b}−\mathrm{1}}{{a}},…….,\frac{{b}+\mathrm{1}}{{a}};\frac{{x}}{{a}^{{a}} }\right) \\ $$$$=\underset{{k}\geqslant\mathrm{0}} {\sum}\frac{{k}!}{\underset{{j}=\mathrm{0}}…

Prove-by-mathematcal-induction-that-1-1-1-2-1-1-2-3-1-1-2-3-n-2n-n-1-

Question Number 15671 by tawa tawa last updated on 12/Jun/17 $$\mathrm{Prove}\:\mathrm{by}\:\mathrm{mathematcal}\:\mathrm{induction}\:\mathrm{that} \\ $$$$\mathrm{1}\:+\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{2}}\:+\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{2}\:+\:\mathrm{3}}\:+\:…\:+\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{2}\:+\:\mathrm{3}\:+\:…\:\mathrm{n}}\:=\:\frac{\mathrm{2n}}{\mathrm{n}\:+\:\mathrm{1}} \\ $$ Answered by icyfalcon999 last updated on 12/Jun/17 $$\left.\mathrm{1}\right)\mathrm{proving}\:\mathrm{that}\:\mathrm{the}\:\mathrm{statement}\:\mathrm{true}\:\mathrm{when}\:\mathrm{n}=\mathrm{1} \\ $$$$\mathrm{R}.\mathrm{H}.\mathrm{S}.=\frac{\mathrm{2}\left(\mathrm{1}\right)}{\mathrm{1}+\mathrm{1}}=\frac{\mathrm{2}}{\mathrm{2}}=\mathrm{1}=\mathrm{L}.\mathrm{H}.\mathrm{S}.…

Question-146157

Question Number 146157 by henderson last updated on 11/Jul/21 Answered by gsk2684 last updated on 11/Jul/21 $$\underset{\frac{\mathrm{1}}{{e}}} {\overset{\lambda} {\int}}\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}−\mathrm{ln}\:{x}\right)\frac{{d}\left(\mathrm{1}−\mathrm{ln}\:{x}\right)}{−\mathrm{1}} \\ $$$$−\frac{\mathrm{1}}{\mathrm{2}}\left[\frac{\left(\mathrm{1}−\mathrm{ln}\:{x}\right)^{\mathrm{2}} }{\mathrm{2}}\right]_{\frac{\mathrm{1}}{{e}}} ^{\lambda} \\ $$$$−\frac{\mathrm{1}}{\mathrm{4}}\left[\left(\mathrm{1}−\mathrm{ln}\:\lambda\right)^{\mathrm{2}}…