Question Number 1791 by RasheedAhmad last updated on 29/Sep/15 $${Simplify}\:: \\ $$$$\left(\mathrm{3}+\mathrm{4}{i}\right)^{\mathrm{3}−\mathrm{4}{i}} \\ $$ Commented by 112358 last updated on 29/Sep/15 $${Define}\:{z}=\left(\mathrm{3}+\mathrm{4}{i}\right)^{\mathrm{3}−\mathrm{4}{i}} \:{and}\:{w}=\mathrm{3}+\mathrm{4}{i}. \\ $$$$\mid{w}\mid=\sqrt{\mathrm{4}^{\mathrm{2}}…
Question Number 1789 by 112358 last updated on 26/Sep/15 $${If}\:\mid{r}\mid\neq\mathrm{1},\:{find}\:{an}\:{expression}\:{for} \\ $$$${T}_{{n}} \left({r}\right),\:{where}\: \\ $$$${T}_{{n}} \left({r}\right)=\mathrm{1}+{r}^{\mathrm{2}} +{r}^{\mathrm{3}} +{r}^{\mathrm{4}} +{r}^{\mathrm{6}} +{r}^{\mathrm{8}} +{r}^{\mathrm{9}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+{r}^{\mathrm{10}} +{r}^{\mathrm{12}} +{r}^{\mathrm{14}}…
Question Number 1784 by 112358 last updated on 25/Sep/15 $${Find}\:{the}\:{sum}\:{of}\:{the}\:{series}\: \\ $$$$\mathrm{1}^{\mathrm{3}} −\mathrm{2}^{\mathrm{3}} +\mathrm{3}^{\mathrm{3}} −\mathrm{4}^{\mathrm{3}} +…+\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{3}} \:. \\ $$ Answered by Rasheed Soomro last updated…
Question Number 1777 by 112358 last updated on 22/Sep/15 $${The}\:{n}\:{positive}\:{numbers}\:{x}_{\mathrm{1}} ,{x}_{\mathrm{2}} ,…{x}_{{n}} \\ $$$${where}\:{n}\geqslant\mathrm{3},\:{satisfy}\: \\ $$$${x}_{\mathrm{1}} =\mathrm{1}+\frac{\mathrm{1}}{{x}_{\mathrm{2}} },{x}_{\mathrm{2}} =\mathrm{1}+\frac{\mathrm{1}}{{x}_{\mathrm{3}} },\:…\:,\:{x}_{{n}−\mathrm{1}} =\mathrm{1}+\frac{\mathrm{1}}{{x}_{{n}} } \\ $$$${and}\:{x}_{{n}} =\mathrm{1}+\frac{\mathrm{1}}{{x}_{\mathrm{1}}…
Question Number 1752 by Rasheed Ahmad last updated on 14/Sep/15 $${Prove}\:{that}: \\ $$$$\left(−{x}\right)\left(−{y}\right)={xy} \\ $$ Commented by 123456 last updated on 15/Sep/15 $${ax}+{bx}=\left({a}+{b}\right){x} \\ $$$$−{x}=−\mathrm{1}×{x}…
Question Number 1748 by Rasheed Ahmad last updated on 14/Sep/15 $${Why}\:\:\:{a}^{\mathrm{0}} =\mathrm{1}\:{and}\:\:{a}^{−\mathrm{3}} =\frac{\mathrm{1}}{{a}^{\mathrm{3}} } \\ $$ Answered by 123456 last updated on 14/Sep/15 $$\frac{{a}^{{x}} }{{a}^{{y}}…
Question Number 1738 by Rasheed Sindhi last updated on 08/Sep/15 $${Let}\:\boldsymbol{\mathrm{a}}\:{belongs}\:{to}\:{an}\:{interval}\:\boldsymbol{\mathrm{A}},\boldsymbol{\mathrm{k}}\:{is}\:{aconstant}\:{such}\:{that}\: \\ $$$$\boldsymbol{\mathrm{k}}\in\mathbb{R}^{+} \:{and}\:\boldsymbol{\mathrm{k}}<\boldsymbol{\mathrm{a}}\:. \\ $$$${Find}\:{out}\:\boldsymbol{\mathrm{A}}\:{in}\:{case}: \\ $$$$\:\:\:\:\:\:\:\:\left(\:\boldsymbol{\mathrm{a}}−\boldsymbol{\mathrm{k}}\right)^{\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{k}}} \geqslant\left(\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{k}}\right)^{\boldsymbol{\mathrm{a}}−\boldsymbol{\mathrm{k}}} \\ $$ Terms of Service Privacy…
Question Number 1729 by Rasheed Ahmad last updated on 04/Sep/15 $${Prove}/{disprove}/{prove}\:{for}\:{an} \\ $$$${interval}\:{as}\:{the}\:{case}\:{may}\:{be}: \\ $$$$\left({x}!\right)^{\frac{\mathrm{1}}{{x}}} \:\overset{?} {<}\:\left\{\left({x}+\mathrm{1}\right)!\right\}^{\frac{\mathrm{1}}{{x}+\mathrm{1}}} \:\:,\:{x}\in\mathbb{N}\:\left[{x}\neq\mathrm{0}\right] \\ $$$$\left({Generalization}\:{of}\:{Q}\:\mathrm{1700}\right) \\ $$ Commented by 123456…
Question Number 1734 by Rasheed Ahmad last updated on 07/Sep/15 $$\boldsymbol{\mathrm{a}}\:\:{belongs}\:{to}\:{an}\:{interval}\:\boldsymbol{\mathrm{A}}\:{and} \\ $$$$\boldsymbol{\mathrm{k}}\in\mathbb{R}^{+} \:\:{is}\:{a}\:{constant}.\:{Determine} \\ $$$$\boldsymbol{\mathrm{A}}\:{in}\:{case} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mid\boldsymbol{\mathrm{a}}−\boldsymbol{\mathrm{k}}\mid^{\mid\:\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{k}}\:\mid} \geqslant\:\mid\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{k}}\mid^{\mid\boldsymbol{\mathrm{a}}−\boldsymbol{\mathrm{k}}\mid} \: \\ $$$$ \\ $$ Terms…
Question Number 1724 by Hasan Mohamed last updated on 03/Sep/15 $$ \\ $$ Answered by 123456 last updated on 04/Sep/15 $${f}\left({x}\right)={x}!=\Gamma\left({x}+\mathrm{1}\right) \\ $$$${f}'\left({x}\right)={x}!\psi\left({x}+\mathrm{1}\right) \\ $$$$\psi\left({x}\right)=\frac{{d}}{{dx}}\mathrm{ln}\:\Gamma\left({x}\right)…