Question Number 83242 by peter frank last updated on 29/Feb/20 Commented by john santu last updated on 29/Feb/20 $$\left(\mathrm{4b}\right)\:\underset{\mathrm{r}\:=\:\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\mathrm{r}\left(\mathrm{r}+\mathrm{1}\right)\:=\:\underset{\mathrm{r}\:=\:\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\mathrm{r}^{\mathrm{2}} \:+\:\underset{\mathrm{r}\:=\:\mathrm{1}} {\overset{\mathrm{n}}…
Question Number 83235 by peter frank last updated on 28/Feb/20 Commented by peter frank last updated on 28/Feb/20 $${c}\:{help} \\ $$ Commented by john santu…
Question Number 148753 by Jonathanwaweh last updated on 30/Jul/21 Answered by mathmax by abdo last updated on 31/Jul/21 $$\mathrm{A}_{\mathrm{n}} =\prod_{\mathrm{k}=\mathrm{2}} ^{\mathrm{n}} \:\mathrm{e}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{k}^{\mathrm{2}} }\right)^{\mathrm{k}^{\mathrm{2}} } \:\Rightarrow\mathrm{A}_{\mathrm{n}}…
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Question Number 83188 by john santu last updated on 28/Feb/20 $$\mathrm{If}\:\underset{\mathrm{a}\:=\:\mathrm{0}} {\overset{\mathrm{n}−\mathrm{1}} {\sum}}\:\left(\mathrm{2a}+\mathrm{1}\right)\mathrm{x}^{\mathrm{2}} +\left(\mathrm{n}^{\mathrm{2}} +\mathrm{4n}−\mathrm{5}\right)\mathrm{x}+\mathrm{16} \\ $$$$=\:\mathrm{0}\:\mathrm{is}\:\mathrm{a}\:\mathrm{perfect}\:\mathrm{square}\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\mathrm{n}\:\in\:\mathbb{Z}^{+} \:.\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\mathrm{x}\:+\mathrm{n}\:?\: \\ $$ Answered…
Question Number 83144 by M±th+et£s last updated on 28/Feb/20 $${show}\:{that} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left(\mathrm{2}{n}−\mathrm{1}\right)\left(\mathrm{3}{n}−\mathrm{1}\right)}=\frac{\mathrm{1}}{\mathrm{6}}\left(\sqrt{\mathrm{3}}\:\pi−\mathrm{9}{log}\left(\mathrm{3}\right)+{log}\left(\mathrm{4096}\right)\right) \\ $$ Answered by mind is power last updated on 28/Feb/20…
Question Number 148677 by nadovic last updated on 30/Jul/21 $$\:\:\mathrm{A}\:\mathrm{series}\:\mathrm{of}\:\mathrm{natural}\:\mathrm{numbers}\:\mathrm{are}\: \\ $$$$\:\:\mathrm{grouped}\:\mathrm{as}\:\mathrm{1}+\left(\mathrm{2}+\mathrm{3}\right)+\left(\mathrm{4}+\mathrm{5}+\mathrm{6}\right)+… \\ $$$$\:\:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:{rth}\:\mathrm{group}\:\mathrm{contains}\:{r}\: \\ $$$$\:\:\mathrm{terms}.\:\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the} \\ $$$$\:\:\mathrm{numbers}\:\mathrm{in}\:\mathrm{the}\:\left(\mathrm{2}{r}−\mathrm{1}\right){th}\:\mathrm{group}\:\mathrm{is} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{r}}^{\mathrm{4}} −\left(\boldsymbol{{r}}−\mathrm{1}\right)^{\mathrm{4}} . \\ $$ Answered…
Question Number 17595 by virus last updated on 08/Jul/17 $${Let}\:{a},{b},{c}\:{be}\:{the}\:{posetive}\:{integer}\:{such}\:{that}\: \\ $$$${b}/{a}\:{is}\:{an}\:{also}\:{integer}\:{if}\:{a},{b},{c}\:{are}\:{in}\:{GP}\:{and}\: \\ $$$${AM}\:{of}\:{a},{b},{c}\:{is}\left({b}+\mathrm{2}\right)\:{then}\:{find}\:{the}\:{value}\:{of} \\ $$$$\left({a}^{\mathrm{2}} +{a}−\mathrm{14}\right)/\left({a}+\mathrm{1}\right) \\ $$ Commented by virus last updated on…
Question Number 148559 by Jonathanwaweh last updated on 29/Jul/21 Answered by Kamel last updated on 29/Jul/21 $${a}=\mathrm{3}{k}+{r},{b}=\mathrm{3}{k}'+{r}'\:\mathrm{0}\leqslant{r}<\mathrm{3},\:\mathrm{0}\leqslant{r}'<\mathrm{3}. \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} =\mathrm{3}{c}=\mathrm{9}\left({k}^{\mathrm{2}} +{k}'^{\mathrm{2}} \right)+\mathrm{6}\left({kr}+{k}'{r}'\right)+{r}^{\mathrm{2}} +{r}'^{\mathrm{2}} \\…
Question Number 82995 by M±th+et£s last updated on 26/Feb/20 Commented by mr W last updated on 26/Feb/20 $${a}_{\mathrm{1000}} =\frac{\mathrm{1}}{\mathrm{499501}}\:? \\ $$ Answered by mind is…