Question Number 5742 by Rasheed Soomro last updated on 26/May/16 $$\bullet{Determine}\:{S} \\ $$$${S}={a}^{\mathrm{2}} +{ar}\left({a}+{r}\right)+{ar}^{\mathrm{2}} \left({a}+\mathrm{2}{r}\right)+…+{ar}^{{n}−\mathrm{1}} \left\{{a}+\left({n}−\mathrm{1}\right)\:{r}\right\}. \\ $$$$ \\ $$ Answered by Yozzii last updated…
Question Number 5744 by Rasheed Soomro last updated on 26/May/16 $${If}\:\:\mathrm{0}<\:{r}<\mathrm{1},\:{is}\:{S}\:\:{convergent}\:{in}\:{the}\:{following}\:? \\ $$$${S}={a}^{\mathrm{2}} +{ar}\left({a}+{r}\right)+{ar}^{\mathrm{2}} \left({a}+\mathrm{2}{r}\right)+….. \\ $$$${Determine}\:{S}\:\:\:{if}\:\:{it}'{s}\:{convergent}. \\ $$ Commented by FilupSmith last updated on…
Question Number 5723 by Rasheed Soomro last updated on 25/May/16 $$\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{8}}+….+\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{n}} }=\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{n}} } \\ $$$$\mathrm{P}\:\:\:\mathrm{r}\:\:\:\mathrm{o}\:\:\:\mathrm{v}\:\:\:\mathrm{e}\:\mathrm{the}\:\mathrm{above}\:\mathrm{for}\:\mathrm{integral}\:\mathrm{n}\geqslant\mathrm{1}. \\ $$ Commented by FilupSmith last updated on 25/May/16 $$\boldsymbol{\mathrm{LHS}}={S}=\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{8}}+…+\frac{\mathrm{1}}{\mathrm{2}^{{n}}…
Question Number 136793 by JulioCesar last updated on 26/Mar/21 Answered by Dwaipayan Shikari last updated on 26/Mar/21 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{{x}^{{sin}\left({ax}\right)} }{{x}^{{tan}\left({bx}\right)} }\right)={y} \\ $$$$\Rightarrow\left({sin}\left({ax}\right)−{tan}\left({bx}\right)\right){log}\left({x}\right)={log}\left({y}\right) \\ $$$$\Rightarrow\left({acos}\left({ax}\right)−{bsec}^{\mathrm{2}}…
Question Number 5722 by Rasheed Soomro last updated on 25/May/16 $$\mathrm{Prove}\:\mathrm{by}\:\boldsymbol{\mathrm{mathematical}}\:\boldsymbol{\mathrm{induction}} \\ $$$$\mathrm{that}\:\mathrm{tbe}\:\mathrm{following}\:\mathrm{formula}\:\mathrm{is}\:\mathrm{correct} \\ $$$$\mathrm{for}\:\mathrm{all}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{n}: \\ $$$$\begin{pmatrix}{\mathrm{2}}\\{\mathrm{2}}\end{pmatrix}\:+\begin{pmatrix}{\mathrm{3}}\\{\mathrm{2}}\end{pmatrix}\:+\begin{pmatrix}{\mathrm{4}}\\{\mathrm{2}}\end{pmatrix}\:+…+\begin{pmatrix}{\mathrm{n}+\mathrm{1}}\\{\:\:\:\mathrm{2}}\end{pmatrix}\:=\begin{pmatrix}{\mathrm{n}+\mathrm{2}}\\{\:\:\:\mathrm{3}}\end{pmatrix} \\ $$ Commented by Yozzii last updated on…
Question Number 5712 by sanusihammed last updated on 24/May/16 $${Find}\:{the}\:{value}\:{of}\:{x}\: \\ $$$$ \\ $$$$\mathrm{2}^{{x}} \:=\:\mathrm{4}{x} \\ $$$$ \\ $$$${workings}\:{is}\:{needed}\:{please}. \\ $$ Commented by Yozzii last…
Question Number 71242 by TawaTawa last updated on 13/Oct/19 $$\mathrm{Given}:\:\:\:\frac{\mathrm{a}}{\mathrm{b}}\:+\:\frac{\mathrm{c}}{\mathrm{d}}\:\:=\:\:\frac{\mathrm{b}}{\mathrm{a}}\:+\:\frac{\mathrm{d}}{\mathrm{c}} \\ $$$$\mathrm{Show}\:\mathrm{that},\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{a}^{\mathrm{2}} }{\mathrm{b}^{\mathrm{2}} }\:−\:\frac{\mathrm{c}^{\mathrm{2}} }{\mathrm{d}^{\mathrm{2}} }\:\:=\:\:\frac{\mathrm{b}^{\mathrm{2}} }{\mathrm{a}^{\mathrm{2}} }\:−\:\frac{\mathrm{d}^{\mathrm{2}} }{\mathrm{c}^{\mathrm{2}} } \\ $$ Answered by MJS…
Question Number 5685 by sanusihammed last updated on 24/May/16 $${Find}\:{the}\:{value}\:{of}\:{x}\:. \\ $$$$ \\ $$$$\mathrm{9}^{{x}} \:=\:\mathrm{6}^{{x}} \:+\:\mathrm{4}^{{x}} \\ $$$$ \\ $$$${Please}\:{help}. \\ $$ Answered by prakash…
Question Number 5684 by sanusihammed last updated on 24/May/16 $${Find}\:{the}\:{value}\:{of}\:{x}\:.\: \\ $$$$ \\ $$$${x}^{\left({x}\:+\:\mathrm{2}\right)} \:=\:\left({x}\:+\:\mathrm{2}\right)^{{x}} \\ $$$$ \\ $$$${Thanks}\:{for}\:{your}\:{help}. \\ $$ Commented by prakash jain…
Question Number 136749 by EDWIN88 last updated on 25/Mar/21 $$\mathrm{Given}\:\mathrm{system}\:\mathrm{equation}\: \\ $$$$\:\begin{cases}{\mathrm{x}^{\mathrm{2}} +\mathrm{3xy}+\mathrm{y}^{\mathrm{2}} +\mathrm{1}=\mathrm{0}}\\{\mathrm{x}^{\mathrm{3}} +\mathrm{y}^{\mathrm{3}} −\mathrm{7}=\mathrm{0}}\end{cases}\:\mathrm{has}\:\mathrm{solution}\: \\ $$$$\left(\mathrm{x}_{\mathrm{1}} ,\mathrm{y}_{\mathrm{1}} \right)\:\&\left(\mathrm{x}_{\mathrm{2}} ,\mathrm{y}_{\mathrm{2}} \right)\:\mathrm{for}\:\mathrm{x},\mathrm{y}\in\mathbb{R}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{value} \\ $$$$\mathrm{of}\:\mathrm{x}_{\mathrm{1}} ^{\mathrm{2}}…