Question Number 142710 by loveineq last updated on 04/Jun/21 $$\mathrm{Let}\:{k}\:\mathrm{be}\:\mathrm{non}-\mathrm{negative}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{and}\:{n}\:\in\:\mathrm{N}^{+} \geqslant\mathrm{1}.\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\frac{\mathrm{4}−{k}}{\mathrm{4}+{k}}\right)\left(\frac{\mathrm{12}−{k}}{\mathrm{12}+{k}}\right)…\left[\frac{\mathrm{2}{n}\left({n}+\mathrm{1}\right)−{k}}{\mathrm{2}{n}\left({n}+\mathrm{1}\right)+{k}}\right]\:\leqslant\:\frac{{n}+{k}+\mathrm{1}}{\left({n}+\mathrm{1}\right)\left({k}+\mathrm{1}\right)} \\ $$$$ \\ $$ Answered by loveineq last updated on…
Question Number 11633 by Joel576 last updated on 29/Mar/17 $$\mathrm{A}\:\mathrm{geometric}\:\mathrm{sequence}\:\mathrm{with}\:{n}\:\mathrm{terms}\: \\ $$$${a}_{\mathrm{1}} ,\:{a}_{\mathrm{2}} ,\:{a}_{\mathrm{3}} ,\:…,\:{a}_{{n}} \:\mathrm{which}\:\mathrm{has}\:{a}_{\mathrm{1}} \:.\:{a}_{{n}} \:=\:\mathrm{3} \\ $$$$\mathrm{If}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of}\:\mathrm{all}\:{n}\:\mathrm{terms}\:=\:{a}_{\mathrm{1}} {a}_{\mathrm{2}} {a}_{\mathrm{3}} …{a}_{{n}} =\:\mathrm{59049} \\…
Question Number 11632 by Joel576 last updated on 29/Mar/17 Answered by sandy_suhendra last updated on 29/Mar/17 Commented by sandy_suhendra last updated on 29/Mar/17 $$\mathrm{Area}\:\Delta\mathrm{AOB}\:+\:\mathrm{Area}\:\Delta\:\mathrm{BPC} \\…
Question Number 77164 by Maclaurin Stickker last updated on 03/Jan/20 Commented by Maclaurin Stickker last updated on 03/Jan/20 $${I}\:{got}\:{x}=\mathrm{5}.\:{Is}\:{my}\:{answer}\:{correct}? \\ $$ Commented by mr W…
Question Number 11628 by uni last updated on 29/Mar/17 $$\mathrm{p}\left(\mathrm{x}−\mathrm{1}\right)+\mathrm{p}\left(\mathrm{x}+\mathrm{1}\right)=\mathrm{4x}^{\mathrm{2}} −\mathrm{2x}+\mathrm{10} \\ $$$$\mathrm{p}\left(\mathrm{x}\right)=? \\ $$ Answered by sandy_suhendra last updated on 29/Mar/17 $$\mathrm{let}\:\mathrm{p}\left(\mathrm{x}\right)=\mathrm{ax}^{\mathrm{2}} +\mathrm{bx}+\mathrm{c} \\…
Question Number 77160 by Chi Mes Try last updated on 03/Jan/20 Commented by Chi Mes Try last updated on 03/Jan/20 $${please}\:{help}\:{out} \\ $$ Answered by…
Question Number 142698 by Willson last updated on 04/Jun/21 $$\mathrm{Prove}\:\mathrm{that}\:\forall\mathrm{n}\in\mathbb{N}^{\ast} \\ $$$$\:\:\:\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}−\mathrm{1}} {\prod}}\mathrm{sin}\left(\frac{\mathrm{k}\pi}{\mathrm{2n}}\right)\:=\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}−\mathrm{1}} {\prod}}\mathrm{cos}\left(\frac{\mathrm{k}\pi}{\mathrm{2n}}\right) \\ $$ Answered by Ar Brandon last updated on…
Question Number 77158 by TawaTawa last updated on 03/Jan/20 $$\int_{−\mathrm{2}} ^{\:\mathrm{2}} \:\left(\mathrm{x}^{\mathrm{3}} \:\mathrm{cos}\frac{\mathrm{x}}{\mathrm{2}}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\right)\sqrt{\mathrm{4}\:−\:\mathrm{x}^{\mathrm{2}} }\:\:\mathrm{dx} \\ $$ Commented by TawaTawa last updated on 03/Jan/20 $$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\…
Question Number 11621 by Nayon last updated on 29/Mar/17 $${Evaluate}\:\:\int\frac{{ax}+{b}}{{cx}+{d}}{dx} \\ $$ Answered by mrW1 last updated on 29/Mar/17 $$=\int\frac{\frac{{a}}{{c}}\left({cx}+{d}\right)+\left({b}−\frac{{ad}}{{c}}\right)}{{cx}+{d}}{dx} \\ $$$$=\frac{{a}}{{c}}\int{dx}+\left({b}−\frac{{ad}}{{c}}\right)×\frac{\mathrm{1}}{{c}}\int\frac{{d}\left({cx}+{d}\right)}{{cx}+{d}} \\ $$$$=\frac{{ax}}{{c}}+\frac{{bc}−{ad}}{{c}^{\mathrm{2}} }×\mathrm{ln}\:\left({cx}+{d}\right)+{C}…
Question Number 142689 by rs4089 last updated on 04/Jun/21 Answered by Dwaipayan Shikari last updated on 04/Jun/21 $$\int_{\mathrm{0}} ^{\infty} \frac{{sin}\left({x}\right)}{{x}}{f}\left({x}\right){dx}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {f}\left({x}\right){dx}\:\:\:\:{f}\left({x}\pm\pi\right)={f}\left({x}\right) \\ $$$${let}\:{f}\left({x}\right)={sin}^{\mathrm{2}{n}−\mathrm{2}} {x}\:\:\:\:\:\:…