Question Number 37692 by kunal1234523 last updated on 16/Jun/18 $$\mathrm{2}+\mathrm{6}+\mathrm{12}+\mathrm{20}+\mathrm{30}+\mathrm{42}+………+\mathrm{n}=\frac{\mathrm{1}}{\mathrm{3}}\left(\mathrm{n}\right)\left(\mathrm{n}−\mathrm{1}\right)\left(\mathrm{n}−\mathrm{2}\right) \\ $$$$\mathrm{is}\:\mathrm{this}\:\mathrm{true}.\:\mathrm{if}\:\mathrm{yes}\:\mathrm{so}\:\mathrm{please}\:\mathrm{derive}\:\mathrm{it}\:\mathrm{from}\:\mathrm{L}.\mathrm{H}.\mathrm{S} \\ $$$$\mathrm{hey}\:\mathrm{I}\:\mathrm{just}\:\mathrm{noticed}\:\mathrm{I}\:\mathrm{got}\:\mathrm{15yrs}\:\mathrm{and}\:\:\mathrm{5}\:\mathrm{months}\:\mathrm{older} \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 16/Jun/18 $${s}=\mathrm{2}+\mathrm{6}+\mathrm{12}+\mathrm{20}+\mathrm{30}+\mathrm{42}+…+{T}_{{n}−\mathrm{1}} +{T}_{{n}}…
Question Number 103219 by Dwaipayan Shikari last updated on 13/Jul/20 $$\mathrm{1}+\mathrm{1}+\mathrm{1}+\mathrm{1}+\mathrm{1}+\mathrm{1}+\mathrm{1}+….={S}_{{n}} \\ $$$${S}_{{n}} =\mathrm{1}+\mathrm{1}+\mathrm{1}+\mathrm{1}+\mathrm{1}+\mathrm{1}+\mathrm{1}+…. \\ $$$$\mathrm{2}{S}_{{n}} =\:\:\:\:\mathrm{2}\:+\:\:\:\:\:\:\mathrm{2}\:\:\:+\:\:\:\:\:\mathrm{2}+……. \\ $$$$……….\:{subtracting} \\ $$$$−{S}_{{n}} =\mathrm{1}−\mathrm{1}+\mathrm{1}−\mathrm{1}+\mathrm{1}−\mathrm{1}+\mathrm{1}−\mathrm{1}+\mathrm{1}−\mathrm{1}+…. \\ $$$$−{S}_{{n}} =\frac{\mathrm{1}}{\mathrm{2}}…
Question Number 103209 by nimnim last updated on 13/Jul/20 Answered by bramlex last updated on 13/Jul/20 $$\mathrm{n}\:=\:\mathrm{7k}+\mathrm{1}\:…\left(\mathrm{1}\right)×\mathrm{6} \\ $$$$\mathrm{n}=\:\mathrm{6p}+\mathrm{3}\:…\left(\mathrm{2}\right)×\mathrm{7} \\ $$$$\mathrm{n}\:=\:\mathrm{5q}\:+\mathrm{2}…\left(\mathrm{3}\right) \\ $$$$\Rightarrow\:\left(\mathrm{2}\right)−\left(\mathrm{1}\right) \\ $$$$\Rightarrow\mathrm{n}\:=\:\mathrm{42}\left(\mathrm{p}−\mathrm{k}\right)+\mathrm{15}\:…\left(\mathrm{4}\right)…
Question Number 168744 by LEKOUMA last updated on 16/Apr/22 $${Resolve} \\ $$$$\left.\mathrm{1}\right)\:\int\frac{\sqrt{\mathrm{1}+\mathrm{cos}\:{x}}}{\mathrm{sin}\:{x}}{dx} \\ $$$$\left.\mathrm{2}\right)\:\int\frac{{dx}}{\mathrm{1}+\sqrt[{\mathrm{3}}]{{x}+\mathrm{1}}} \\ $$$$\left.\mathrm{3}\right)\:\int\frac{{x}\mathrm{tan}\:{x}}{\mathrm{cos}\:^{\mathrm{4}} {x}}{dx} \\ $$$$\left.\mathrm{4}\right)\:\int\frac{{dx}}{\mathrm{1}+\sqrt{{x}}+\sqrt{\mathrm{1}+{x}}} \\ $$ Answered by bobhans last…
Question Number 168742 by LEKOUMA last updated on 16/Apr/22 $${Resolve} \\ $$$${y}={xy}'+{a}\sqrt{\mathrm{1}+\left(\mathrm{y}'\right)^{\mathrm{2}} } \\ $$ Answered by mindispower last updated on 19/Apr/22 $$\Rightarrow \\ $$$${y}'={xy}''+{y}'+{a}\frac{\mathrm{2}{y}'{y}''}{\mathrm{2}\sqrt{\mathrm{1}+{y}'^{\mathrm{2}}…
Question Number 37663 by Rio Mike last updated on 16/Jun/18 $$\mathrm{show}\:\mathrm{that}\:\frac{{sin}\mathrm{2}{A}}{\mathrm{1}+{cos}\mathrm{2}{A}}\:=\:\mathrm{tanA}. \\ $$ Answered by Joel579 last updated on 16/Jun/18 $$\frac{\mathrm{2}\:\mathrm{sin}\:{x}\:\mathrm{cos}\:{x}}{\mathrm{1}\:+\:\left(\mathrm{2cos}^{\mathrm{2}} \:{x}\:−\:\mathrm{1}\right)} \\ $$$$=\:\frac{\mathrm{2}\:\mathrm{sin}\:{x}\:\mathrm{cos}\:{x}}{\mathrm{2}\:\mathrm{cos}^{\mathrm{2}} \:{x}}…
Question Number 37664 by Rio Mike last updated on 16/Jun/18 $$\mathrm{find}\:\int\left(\mathrm{1}\:+\:{sinx}\right){dx} \\ $$ Answered by Joel579 last updated on 16/Jun/18 $$\int\:\mathrm{1}\:{dx}\:+\:\int\:\mathrm{sin}\:{x}\:{dx} \\ $$$$=\:{x}\:−\:\mathrm{cos}\:{x}\:+\:{C} \\ $$…
Question Number 37662 by Rio Mike last updated on 16/Jun/18 $$\:\mathrm{Given}\:\mathrm{that}\:{y}={x}^{\mathrm{2}} {cosx}, \\ $$$$\mathrm{find}\:\frac{\boldsymbol{{dy}}}{\boldsymbol{{dx}}\:},\:\mathrm{simplifying}\:\mathrm{your}\:\mathrm{answer} \\ $$$$\mathrm{as}\:\mathrm{far}\:\mathrm{as}\:\mathrm{posible} \\ $$ Answered by Joel579 last updated on 16/Jun/18…
Question Number 37661 by Rio Mike last updated on 16/Jun/18 $$\mathrm{Given}\:\mathrm{the}\:\mathrm{lines}\: \\ $$$${l}_{\mathrm{1}} :\:\mathrm{r}=\:−\mathrm{5}{i}\:+\:\mathrm{2}{j}\:+\:{s}\left(\mathrm{3}{i}−{j}\right) \\ $$$${l}_{\mathrm{2}} :\:{r}=\:−\mathrm{2}{i}+{j}\:+\:{t}\left(\mathrm{2}{i}+{j}\right) \\ $$$${F}\mathrm{ind}\:\mathrm{the}\:\mathrm{cosine}\:\mathrm{of}\:\mathrm{the}\:\mathrm{angle} \\ $$$$\mathrm{between}\:{l}_{\mathrm{1}} \:{and}\:{l}_{\mathrm{2}} . \\ $$…
Question Number 168723 by LEKOUMA last updated on 16/Apr/22 $${Resolve}\: \\ $$$$\left({x}+\mathrm{5}\right)^{\mathrm{5}} {y}^{''} =\mathrm{1} \\ $$ Commented by greougoury555 last updated on 16/Apr/22 $$\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}}…